Classifying presentations of finite groups -- the case of dicyclic groups

TL;DR

This paper classifies presentations of dicyclic groups up to isomorphism, identifying all equivalence classes with two generators and revealing new presentation types that depend on the group's order and parity.

Contribution

It provides a complete classification of presentations of dicyclic groups with two generators, including new types based on group order and parity, expanding understanding of their structure.

Findings

Finite number of presentation classes for dicyclic groups

Identified all two-generator presentations including new types

Results aid in characterizing group structure and properties

Abstract

The problem of classifying equivalence classes of presentations up to isomorphism of Cayley graphs is considered in this article in the case of dicyclic groups. The number of equivalence classes of presentations is uniformly bounded - it is a "finite presentation type" case. We find all equivalence classes of presentations of dicyclic groups having two generators. For the dicyclic group of order $4n$ apart from the classical presentation with order multiset $\{\{2n,4\}\}$ for all $n$ there are presentations with order multiset $\{\{4,4\}\}$. If $n$ is odd there is an additional presentation having elements with order multiset $\{\{n,4\}\}$. These results may be used in characterizing group structure and properties.

Equations

$$e\to v\to u^2\to u^{2}v\to e.$$

$$G_2=\{v(u^{3}v),uv(u^{3}v),u^{2}v(u^{3}v),(u^{3}v{)}^2\}.$$

$$u^{3}v\to v(u^{3}v)\to uv\to u^{2}v(u^{3}v)\to u^{3}v.$$

$$G_{n-1}=\{v(u^{3}v{)}^{n-2},uv(u^{3}v{)}^{n-2},u^{2}v(u^{3}v{)}^{n-2},(u^{3}v{)}^{n-1}\}.$$

$$G_0=\{e,u,u^2,u^3\},G_j=\{v(u^{3}v{)}^{j-1},uv(u^{3}v{)}^{j-1},u^{2}v(u^{3}v{)}^{j-1},u^{3}v(u^{3}v{)}^{j-1}\}$$

$$(u^{3}v{)}^{n-1}\to u^3\to (uv)(u^{3}v{)}^{n-2}\to u\to (u^{3}v{)}^{n-1},$$

$$(u^{3}v{)}^{n-1}\to u\to (uv)(u^{3}v{)}^{n-2}\to u^3\to (u^{3}v{)}^{n-1}.$$

$${\psi }_i\circ {\phi }_i=i{d}_{D{C}_{4n}},{\phi }_i\circ {\psi }_i=i{d}_{{\Pi }_{4n,i}}.$$

$$\left\{\begin{array}[]{ll}\varphi_{1}(a)=u^{3}v\\ \varphi_{1}(x)=v\\ \end{array}\right.and\ \left\{\begin{array}[]{ll}\psi_{1}(u)=ax\\ \psi_{1}(v)=x\\ \end{array}\right..$$

$$\left\{\begin{array}[]{ll}\varphi_{0}(a)=vu\\ \varphi_{0}(x)=v\\ \end{array}\right.and\ \left\{\begin{array}[]{ll}\psi_{0}(u)=a^{n-1}x\\ \psi_{0}(v)=x\\ \end{array}\right..$$

$$\left\{\begin{array}[]{ll}\varphi(a)=b^{q}y^{2},where\ q=\frac{n+1}{2}\\ \varphi(x)=y\\ \end{array}\right.and\ \left\{\begin{array}[]{ll}\psi(b)=a^{2}\\ \psi(y)=x\\ \end{array}\right..$$

Full text

1 Introduction and outline

Given a group $G$ with a generating sequence $\mathcal{S}$ we define the edge-labeled Cayley graph $\Gamma(G,\mathcal{S})$ in the standard way on the vertex set $G$ with labeled directed edges corresponding to left multiplication by elements of $\mathcal{S}$. Two presentations $\langle\mathcal{S}_{1}|R_{1}\rangle$ and $\langle\mathcal{S}_{2}|R_{2}\rangle$ (and corresponding generating sequences $\mathcal{S}_{1}$,$\mathcal{S}_{2}$) are defined equivalent iff $\Gamma(G,\mathcal{S}_{1})$ and $\Gamma(G,\mathcal{S}_{2})$ are isomorphic as edge-labeled directed graphs, up to edge relabelings.

In this article we solve the problem of finding all equivalence classes of presentations with two generators for a series of finite groups - dicyclic groups. The dicyclic groups are chosen as one of the first cases of this problem for the author because in this case the problem can be called ”of finite presentation type” using the analogy of linear representation theory - the number of equivalence classes of presentations is uniformly bounded for all orders. The main result of the article can be summarized in the following theorem.

Theorem 1**.**

Let $DC_{4n}=\langle a,x|a^{2n}=e,x^{2}=a^{n},x^{-1}ax=a^{-1}\rangle,n\in\mathbb{N},n\geq 2$ (the classical presentation of this group).

If $2|n$ then there are two equivalence classes of minimal presentations with two generators of $DC_{4n}$: the classical presentation and $\Pi_{4n,1}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(u^{3}v)^{n}=e\rangle$. 2. 2.

If $2\not|n$ then there are four equivalence classes of minimal presentations with two generators of $DC_{4n}$: the classical presentation, $\Pi_{4n,1}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(u^{3}v)^{n}=e\rangle$, $\Pi_{4n,0}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(uv)^{n}=e\rangle$ and $\Pi_{4n,n}=\langle b,y|b^{n}=e,y^{4}=e,y^{-1}by=b^{-1}\rangle$.

In group theory one usually works with fixed (classical) presentations. For important series of groups such as symmetric or alternating groups, certain generators and presentations have been accepted as standard ones. Interesting problems of finding generators with given low orders have been solved for symmetric and alternating groups, see [2]. Generators and presentations of simple groups is an active research area, see [3].

Definition of natural equivalence relations and classification of equivalence classes of mathematical objects in any area is a motivated, albeit often auxiliary, problem once these objects are defined. In algebra useful equivalence relations are defined considering changes of generators of algebraic objects.

The problem of defining equivalence relations on sets of presentations and describing all equivalence classes of presentations does not seem to have been clearly formulated and addressed in the literature. In group theory classification of presentations may be related to some general problems of group theory such as classification of groups. This problem is trivial for extreme cases such as cyclic or elementary abelian groups. Other cases may give additional description of groups.

A suitable graph-based technique is introduced. All groups considered in this article are finite.

2 Review

2.1 Sequences

Given sequences $\mathcal{A}_{i}=(a_{i1},...,a_{in_{i}})$, $i\in\{1,...,m\}$ we define their concatenation $\mathcal{A}=\mathcal{A}_{1}...\mathcal{A}_{m}$ in the standard way. We assume that each $\mathcal{A}_{i}$ is a subsequence of $\mathcal{A}$. Given two sequences $\mathcal{A}=(a_{1},...,a_{m})$, $\mathcal{B}=(b_{1},...,b_{m})$, the function $f:\mathcal{A}\rightarrow\mathcal{B}$ is a sequence of assignments $f(a_{i})=b_{i}$. Given a sequence $\mathcal{S}$ we define $Set(\mathcal{S})$ to be the underlying set of $\mathcal{S}$ . We denote union of multisets by $\coprod$. Double curly brackets are used for multisets.

2.2 Group presentations and Cayley graphs

An edge-labeled graph is a quadruple $\Gamma=(V,E,k,w)$, where $k$ is the set of edge labels  and $w:E\rightarrow k$ is an edge-label function, $w(a,b)=w$ means that the edge $a\rightarrow b$, also denoted as the ordered pair $(a,b)$, is given the label $w$, in other notations $a\stackrel{{\scriptstyle w}}{{\rightarrow}}b$, $(a,b)_{w}$. We denote the corresponding undirected edge-labeled graph by $\Gamma_{u}$.

Two edge-labeled graphs $\Gamma_{1}=(V_{1},E_{1},k_{1},w_{1})$ and $\Gamma_{2}=(V_{2},E_{2},k_{2},w_{2})$ are isomorphic ($\Gamma_{1}\simeq\Gamma_{2}$) if there are two bijective functions $f:V_{1}\rightarrow V_{2}$, $\sigma:k_{1}\rightarrow k_{2}$ such that $a\stackrel{{\scriptstyle w}}{{\rightarrow}}b$ iff $f(a)\stackrel{{\scriptstyle\sigma(w)}}{{\rightarrow}}f(b)$, for any pair $a,b$. The graphs $\Gamma_{1}$ and $\Gamma_{2}$ are called undirected-isomorphic ($\Gamma_{1}\simeq_{u}\Gamma_{2}$) if $(\Gamma_{1})_{u}\simeq(\Gamma_{2})_{u}$, as undirected edge-labeled graphs.

Let $G$ be a group. In this article we consider generating sequences instead of traditional generating sets, relations are still considered as sets. Let $\mathcal{S}$ be a sequence of $G$-elements: $\mathcal{S}\in G^{l}$, $l=|\mathcal{S}|$ . We denote $\langle\mathcal{S}\rangle=\langle Set(\mathcal{S})\rangle$ and $S=Set(\mathcal{S})$. Let $E_{S}=\bigcup\limits_{g\in G,s\in S}(g,sg)_{s}$. $w_{S}:E_{S}\rightarrow S$ is defined as follows: $w_{S}(g,sg)=s$. The edge-labeled graph $\Gamma(G,\mathcal{S})=(G,E_{S},S,w_{S})$ is called the Cayley graph of $G$ with respect to the sequence $\mathcal{S}$. If $G=\langle\mathcal{S}\rangle$ then $\Gamma(G,\mathcal{S})$ is connected. For any two group elements $g_{1}\in G,g_{2}\in G$ there is a unique edge-labeled graph automorphism of $\Gamma(G,\mathcal{S})$ sending $g_{1}$ to $g_{2}$. A group automorphism $\varphi:G\rightarrow G$ induces a graph isomorphism (Cayley isomorphism) $\Gamma(G,\mathcal{S})\rightarrow\Gamma(G,\varphi(\mathcal{S}))$. See [4].

2.3 Notations and review of dicyclic groups

In terms of 1 denote $\mathbb{A}_{4n}=\bigcup_{i=0}^{2n-1}a^{i}$ and $\mathbb{X}_{4n}=\bigcup_{j=0}^{2n-1}a^{j}x$. We have that $DC_{4n}=\mathbb{A}_{4n}\cup\mathbb{X}_{4n}$ . We note the following obvious multiplication rules: $(a^{k}x)(a^{m}x)=a^{k-m+n}$, $(a^{k}x)^{-1}=a^{k+n}x$. $DC_{2^{k}}=Q_{2^{k}}$ is called generalized quaternion group, see [5].

3 Main results

3.1 Graph-based equivalence relation of presentations

Definition 2**.**

$G$* - a group, $\mathcal{S}_{i}$ - $G$-generating sequences, $i\in\{1,2\}$, $G=\langle\mathcal{S}_{i}\rangle$, $R_{i}$ - sets of relations between elements of $\mathcal{S}_{i}$. The presentations $\langle\mathcal{S}_{1}|R_{1}\rangle$ and $\langle\mathcal{S}_{2}|R_{2}\rangle$ (and corresponding generating sequences/sets) will be called equivalent* (denoted $\langle\mathcal{S}_{1}|R_{1}\rangle\simeq\langle\mathcal{S}_{2}|R_{2}\rangle$) if $\Gamma(G,\mathcal{S}_{1})\simeq\Gamma(G,\mathcal{S}_{2})$.

Studying any group or a family of groups we may pose and solve the problem of finding all equivalence types of minimal presentations.

Example 3**.**

*The group of minimal cardinality having two nonequivalent presentations of the same number of generators is $\Sigma_{3}$. It can be generated by any two elements of orders $2$ and $3$, or by any two elements of order $2$. Computations show that symmetric and alternating groups have more than one equivalence class of minimal presentations: for $\Sigma_{4}$ there are $5$ classes of minimal presentations with two generators and $9$ classes of minimal presentations with three generators. *

Example 4**.**

We assume it known that for the dihedral group $D_{2n}$, $n\geq 3$, there are two equivalence types of presentations with two elements - $\langle a,x|a^{n}=x^{2}=e,xax=a^{-1}\rangle$ and $\langle u,v|u^{2}=v^{2}=(uv)^{n}=e\rangle$, where $u=a^{k_{1}}x$ and $v=a^{k_{2}}x$ with $GCD(k_{1}-k_{2},n)=1$. It can also be proved by methods of this article. Thus the classification problem for dihedral groups is also of ”finite presentation type”.

A sufficient condition for two presentations to be non-equivalent is nonequality of multisets of generator orders. Given a sequence $\mathcal{S}=(s_{1},..,s_{n})$, $s_{i}\in G$, define $om(\mathcal{S})=\coprod_{i}^{n}Ord(s_{i})$ - the order multiset of $\mathcal{S}$.

Proposition 5**.**

$G$* - a group, $\mathcal{S},\mathcal{T}$ - sequences of $G$-elements. $\Gamma(G,S)\simeq\Gamma(G,T)$ implies $om(\mathcal{S})=om(\mathcal{T})$.*

Proof.

Vertices of oriented loops corresponding to relations $s^{k_{i}}_{i}=e_{G}$, $s_{i}\in Set(\mathcal{S})$, $k_{i}=Ord(s_{i})$, are mapped by graph isomorphisms to vertices of loops corresponding to relations $t^{m_{j}}_{j}=e_{G}$, $t_{j}\in Set(\mathcal{T})$, $m_{j}=Ord(t_{j})$, for some $j$. For each $i$ we must have $k_{i}=m_{j}$, thus a Cayley graph isomorphism defines a function $om(\mathcal{S})\rightarrow om(\mathcal{T})$ which permutes equal elements. If $om(\mathcal{S})\neq om(\mathcal{T})$, then a bijective function with such property is not possible. ∎

Remark 6**.**

Equality of generator order multisets is not a sufficient condition for presentations to be equivalent. The smallest group having at least two non-equivalent presentations with two generators and the same order multiset is $\mathbb{Z}_{3}\times\mathbb{Z}^{2}_{2}$, it has two non-equivalent presentations each with order multiset $\{\{6,6\}\}:=\{\{6^{2}\}\}$ (two elements of order $6$).

Additionally we can define an equivalence relation using isomorphism of undirected edge-labeled Cayley graphs.

Definition 7**.**

$G$, $\mathcal{S}_{i}$ as in Definition 2. The presentations $\langle\mathcal{S}_{1}|R_{1}\rangle$ and $\langle\mathcal{S}_{2}|R_{2}\rangle$ are called undirected-equivalent (denoted $\langle\mathcal{S}_{1}|R_{1}\rangle\simeq_{u}\langle\mathcal{S}_{2}|R_{2}\rangle$) if $\Gamma(G,\mathcal{S}_{1})\simeq_{u}\Gamma(G,\mathcal{S}_{2})$.

Example 8**.**

$A_{4}$* can be generated by two $3$-cycles in two non-equivalent ways: $A_{4}=\langle(1,2,3),(2,4,3)\rangle=\langle(1,2,3),(2,3,4)\rangle$, but these presentations are undirected-equivalent.*

3.2 Minimal generating sequences for dicyclic groups

3.2.1 Generating elements of order $2n$ and $4$

Proposition 9**.**

Consider $DC_{4n}$ as defined in 1. Any presentation with two generators with orders $2n$ and $4$ is equivalent to the classical presentation.

Proof.

$\mathbb{A}_{4n}$ is the only cyclic subgroup of order $2n$. If $\mathbb{A}_{4n}=\langle b\rangle$, then $b=a^{t}$ with $t$ invertible mod $2n$, $DC_{4n}=\langle b,y\rangle$ for any $y=a^{m}x\in\mathbb{X}_{4n}$. $b$ and $y$ satisfy relations $b^{2n}=e$, $b^{n}=(a^{n})^{t}=a^{n}=x^{2}=y^{2}$, $y^{-1}by=b^{-1}$, thus all such presentation are equivalent. ∎

3.2.2 Two generating elements of order $4$

Proposition 10**.**

Consider $DC_{4n}$ as defined in 1. Let $u=a^{k}x$, $v=a^{m}x$. Then $\langle u,v\rangle=DC_{4n}$ iff $GCD(n,k-m)=1$.

Proof.

If $GCD(n,k-m)=1$, then there exist $\alpha,\beta\in\mathbb{Z}$ such than $1=\alpha n+\beta(k-m)$. We have that $u^{2}=a^{n}$ and $u^{-1}v=a^{k-m}$. It follows that $a=u^{2\alpha}(u^{-1}v)^{\beta}\in\langle u,v\rangle$, $x=a^{-k}u\in\langle u,v\rangle$, and thus $\langle u,v\rangle=DC_{4n}$.

Let $GCD(n,k-m)=d>1$. We prove by induction that a proper subset of $DC_{4n}$ is closed under generation by $\{u,v\}$ and contains $\{u,v\}$, and thus $\langle u,v\rangle\neq DC_{4n}$.

We say that $Y\subseteq DC_{4n}$ is $d$-special (and contains $d$-special elements) iff 1) $Y\cap\mathbb{A}_{4n}\subseteq\langle a^{d}\rangle$ and 2) if $a^{k_{1}}x\in Y$ and $a^{k_{2}}x\in Y$, then $k_{1}\equiv k_{2}(mod\ d)$. Note that $d|2n$, $d>1$, implies that a $d$-special set is a proper subset of $DC_{4n}$. Note that inverses of $d$-special elements are $d$-special.

Define $S_{0}=\{u,v\}$, note that $S_{0}$ is $d$-special. Inductive hypothesis - suppose that after $k$ steps (adding products) we generate a $d$-special subset $S_{k}\subseteq DC_{4n}$. We prove that after $k+1$ steps we will get a $d$-special set $S_{k+1}$. So se have to prove that a product of two $d$-special elements is $d$-special: 1) $a^{k_{1}d}a^{k_{2}d}=a^{(k_{1}+k_{2})d}\in\langle a^{d}\rangle$, 2) $a^{kd}(a^{t}x)=a^{kd+t}x$, we have that $t\equiv kd+t(mod\ d)$, 3) $(a^{t}x)a^{kd}=a^{t-kd+n}x$, we have that $t\equiv t-kd+n(mod\ d)$, 4) let $t_{1}\equiv t_{2}(mod\ d)$, then $(a^{t_{1}}x)(a^{t_{2}}x)=a^{t_{1}-t_{2}+n}$, we have that $t_{1}-t_{2}+n\equiv 0(mod\ d)$.

We have proved that from $S_{0}$ we can generate only $d$-special subsets of $DC_{4n}$. Thus $d>1$ implies $\langle u,v\rangle\neq DC_{4n}$. ∎

Corollary 11**.**

Since $Ord(a^{k}x)=4$, for any $k\in\mathbb{Z}$, it follows from Proposition 10 that for any $n\geq 2$ there are presentations of $DC_{4n}$ having two elements of order $4$.

Proposition 12**.**

Consider $DC_{4n}$ as defined in 1.

If $2|n$ then there is one equivalence type of presentations, i.e. let $k_{i}\neq m_{i}$, $\mathcal{S}_{i}=(a^{k_{i}}x,a^{m_{i}}x)$, $i\in\{1,2\}$ $GCD(n,k_{i}-m_{i})=1$, then $\Gamma(G,\mathcal{S}_{1})\simeq\Gamma(G,\mathcal{S}_{2})$. 2. 2.

If $2\not|n$ then there is two equivalence types of presentations, i.e. let $k_{i}\neq m_{i}$, $\mathcal{S}_{i}=(a^{k_{i}}x,a^{m_{i}}x)$, $i\in\{1,2\}$, $GCD(n,k_{i}-m_{i})=1$, then $\Gamma(G,\mathcal{S}_{1})\simeq\Gamma(G,\mathcal{S}_{2})$ iff $k_{1}-m_{1}\equiv k_{2}-m_{2}(mod\ 2)$.

Proof.

1. Let $2|n$, $G=DC_{4n}$. Choose $k\neq m$ such that $GCD(n,k-m)=1$, $u=a^{k}x$, $v=a^{m}x$. Define the generating sequence $\mathcal{S}=(u,v)$. We start constructing $\Gamma(G,\mathcal{S})$ from $e$ in the following steps.

2. Step 0

Apply $u,u^{2},u^{3}$ to $e$, get the set $G_{0}=\{e,u,u^{2},u^{3}\}$. 2. Step 1

Apply first $v$, then $u,u^{2},u^{3}$ to $e$, generate $G_{1}=\{v,uv,u^{2}v,u^{3}v\}$. Find all $v$-edges between $G_{0}$ and $G_{1}$. For all $k,m$ there are $4$ $v$-edges

[TABLE] 3. Step 2

Apply first $v$, then $u,u^{2},u^{3}$ to $u^{3}v$, generate

[TABLE]

Find all $v$-edges between $G_{1}$ and $G_{2}$. For all $k,m$ there are $4$ $v$-edges

[TABLE] 4. …

… 5. Step n-1

Apply first $v$, then $u,u^{2},u^{3}$ to $(u^{3}v)^{n-2}$, generate

[TABLE]

Find all $v$-edges between $G_{n-2}$ and $G_{n-1}$. For all $k,m$ there are $v$-edges $(u^{3}v)^{n-2}\rightarrow v(u^{3}v)^{n-2}\rightarrow uv(u^{3}v)^{n-3}\rightarrow u^{2}v(u^{3}v)^{n-2}\rightarrow u^{3}v$. 6. Step n

Find all $v$-edges between $G_{n-1}$ and $G_{0}$. For all $k,m$ there are $v$-edges $(u^{3}v)^{n-1}\rightarrow u^{3}\rightarrow uv(u^{3}v)^{n-2}\rightarrow u\rightarrow(u^{3}v)^{n-1}$.

We have that

[TABLE]

for $i\in\{1,...,n-1\}$, $G=\cup_{i=0}^{n-1}G_{i}$. There are no other edges. We see that the Cayley graph construction is uniquely determined for all $k,m$.

Let $\mathcal{S}_{i}=(a^{k_{i}}x,a^{m_{i}}x)$, $i\in\{1,2\}$. Define $u_{i}=a^{k_{i}}x$, $v_{i}=a^{m_{i}}x$. Define sequences $\mathcal{G}_{0,i}=(e,u_{i},u^{2}_{i},u^{3}_{i})$, $\mathcal{G}_{j,i}=(v_{i}(u^{3}_{i}v_{i})^{j-1},u_{i}v_{i}(u^{3}_{i}v_{i})^{j-1},u^{2}_{i}v_{i}(u^{3}_{i}v_{i})^{j-1},u^{3}_{i}v_{i}(u^{3}_{i}v_{i})^{j-1})$ for $j\in\{1,...,n-1\}$, $i\in\{1,2\}$.

By the uniqueness of Cayley graph construction it follows that the bijective function $\varphi:G\rightarrow G$, defined by $\varphi(e)=e$, $\varphi(\mathcal{G}_{j,1})=\mathcal{G}_{j,2}$, for each $j\in\{0,...,n-1\}$, is an isomorphism between $\Gamma(G,\mathcal{S}_{1})$ and $\Gamma(G,\mathcal{S}_{2})$ with the edge-relabeling function $\sigma$ such that $\sigma(u_{1})=u_{2}$, $\sigma(v_{1})=v_{2}$.

2. Let $2\not|n$, $G=DC_{4n}$. Choose $k\neq m$ such that $GCD(n,k-m)=1$, $u=a^{k}x$, $v=a^{m}x$, $\mathcal{S}=(u,v)$. Again we start constructing $\Gamma(G,\mathcal{S})$ from $e$.

First $n-1$ steps are the same as in proof of statement 1, the construction is unique.

For the Step n there are 2 possibilities:

if $k-m\equiv 1(mod\ 2)$, then there are $4$ $v$-edges

[TABLE] 2. 2.

if $k-m\equiv 0(mod\ 2)$, then there are $4$ $v$-edges

[TABLE]

There are no other edges.

If $k_{1}-m_{1}\equiv k_{2}-m_{2}(mod\ 2)$, then $\Gamma(G,\mathcal{S}_{1})\simeq\Gamma(G,\mathcal{S}_{2})$ by the same argument as in 1.

Let $k_{1}-m_{1}\not\equiv k_{2}-m_{2}(mod\ 2)$, $S_{i}=\{a^{k_{i}}x,a^{m_{i}}x\}$, $i\in\{1,2\}$. Define $u_{i}=a^{k_{i}}x$, $v_{i}=a^{m_{i}}x$. Suppose that $k_{1}-m_{1}\equiv 1(mod\ 2)$. We show by contradiction that in this case $\Gamma(G,\mathcal{S}_{1})\not\simeq\Gamma(G,\mathcal{S}_{2})$.

By construction $G=\cup_{i=0}^{n-1}G_{1,i}=\cup_{i=0}^{n-1}G_{2,i}$. If $\Gamma(G,\mathcal{S}_{1})\simeq\Gamma(G,\mathcal{S}_{2})$, then there is an isomorphism $\psi:\Gamma(G,\mathcal{S}_{1})\rightarrow\Gamma(G,\mathcal{S}_{2})$ fixing $e$. We show that it is impossible. We have to consider two possible edge-relabeling functions $\sigma$ and $\tau$: $\sigma(u_{1},v_{1})=(u_{2},v_{2})$, $\tau(u_{1},v_{1})=(v_{2},u_{2})$.

Case $\sigma$

We must have $\psi(\mathcal{G}_{1,i})=\psi(\mathcal{G}_{2,i})$, thus $\psi$ is completely determined. We check if $\psi$ maps edges to edges mapping edge labels by $\sigma$. Considering $v$-edges between $G_{i,0}$ and $G_{i,n-1}$ we get a contradiction: it is sufficient to notice that there is a $v$-edge $u\rightarrow(u^{3}v)^{n-1}$ in $\Gamma(G,\mathcal{S}_{1})$, but a $v$-edge $u\rightarrow(uv)(u^{3}v)^{n-2}$ in $\Gamma(G,\mathcal{S}_{2})$.

Case $\tau$

We generate $G$ in the same way as above interchanging generators in the generating sequence. Again we get two possibilities in the last step. By the same argument we have $\Gamma(G,\mathcal{S}_{1})\not\simeq\Gamma(G,\mathcal{S}_{2})$. ∎

Remark 13**.**

Note that the underlying undirected graphs of both non-isomorphic directed graphs in statement 2, 12, are isomorphic as edge-labeled undirected graphs. Thus there is one equivalence class of presentations with two generators of order $4$ in the sense of Definition 7.

Proposition 14**.**

If $2|n$ then $DC_{4n}$ is isomorphic to $\Pi_{4n,1}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(u^{3}v)^{n}=e\rangle$. 2. 2.

If $2\not|n$ then $DC_{4n}$ is isomorphic to $\Pi_{4n,1}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(u^{3}v)^{n}=e\rangle$ and $\Pi_{4n,0}=\langle u,v|u^{2}=v^{2},u^{4}=u^{2}(uv)^{n}=e\rangle$.

Proof.

It is sufficient to exhibit group morphisms $\varphi_{i}:DC_{4n}\rightarrow\Pi_{4n,i}$ and $\psi_{i}:\Pi_{4n,i}\rightarrow DC_{4n}$ such that

[TABLE]

1. Define $\varphi_{1}$ and $\psi_{1}$ on the generators:

[TABLE]

2. Additionally we define $\varphi_{0}$ and $\psi_{0}$ on the generators:

[TABLE]

We check that $\varphi_{i}$ and $\psi_{i}$ can be extended to group morphisms and 1 hold. For $\varphi_{0}$ and $\psi_{0}$ we take into account that $n$ is odd. ∎

Example 15**.**

Any presentation of $DC_{12}$ having two elements of order $4$ is equivalent either to presentation

$\Pi_{12,1}=\langle u,v|u^{4}=e,u^{2}=v^{2},u^{2}(u^{3}v)^{3}\rangle$, or $\langle ax,x\rangle$, or 2. 2)

$\Pi_{12,0}=\langle u,v|u^{4}=e,u^{2}=v^{2},u^{2}(uv)^{3}\rangle\rangle$, $\langle a^{2}x,x\rangle$.

$\Gamma(DC_{12},(ax,x))$* is shown in Fig.1. continuous arrows mean left multiplication by $u=ax$ and dotted arrows mean left multiplication by $v=x$.*

\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{3}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{2}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{uv\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(u^{3}v)^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{u^{2}vu^{3}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{vu^{3}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{uvu^{3}v\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} **

Fig.1. - the graph $\Gamma(DC_{12},(ax,x))$.

3.2.3 Generating elements of order $n$ and $4$

Proposition 16**.**

Consider $DC_{4n}$ as defined in 1. If $DC_{4n}=\langle b,y\rangle$, $b\in\mathbb{A}_{4n}$, $y\in\mathbb{X}_{4n}$, then there are two possibilities:

$Ord(b)=2n$* or* 2. 2.

$2\not|n$* and $Ord(b)=n$.*

Proof.

If $\langle v,z\rangle=DC_{4n}$ then $v\in\mathbb{X}_{4n}$ or $z\in\mathbb{X}_{4n}$. Let $z=a^{m}x\in\mathbb{X}_{4n}$. The case $v\in\mathbb{X}_{4n}$ has already been discussed. Let $v=a^{k}$. By an inductive argument similar to that in the proof of 10, we can show that $\langle a^{k},a^{m}x\rangle=DC_{4n}$ iff $GCD(n,k)=1$.

For $GCD(n,k)=1$ there are two possibilities: $GCD(2n,k)=1$ or $GCD(2n,k)=2$ and $2\not|n$ . The case $GCD(2n,k)=1$ has been discussed in subsection 3.2.1.

If $GCD(2n,k)=2$, then $k=2^{t}q^{\prime}$, $GCD(n,q^{\prime})=1$. If $Ord(a^{k})=r$, then $2^{t}q^{\prime}r\equiv 0(mod\ 2n)$. It follows that $n|r$, thus $Ord(a^{k})=n$ in this case. ∎

Proposition 17**.**

Consider $DC_{4n}$ as defined in 1, $2\not|n$, $n\geq 3$. Then

$DC_{4n}=\langle a^{2},x\rangle$, 2. 2.

$DC_{4n}\simeq\Pi_{4n,n}$, where $\Pi_{4n,n}=\langle b,y|b^{n}=e,y^{4}=e,y^{-1}by=b^{-1}\rangle$, 3. 3.

If $\mathcal{S}=(b,y)$, $b\in\mathbb{A}_{4n},y\in\mathbb{X}_{4n}$, $Ord(b)=n$, $Ord(y)=4$, then $\Gamma(DC_{4n},\mathcal{S})\simeq\Gamma(DC_{4n},(a^{2},x))$.

Proof.

1. $2\not|n$ implies $a^{n+1}\in\langle a^{2}\rangle$. $a^{n}=x^{2}$ implies $a\in\langle a^{2},x\rangle$ and $\langle a^{2},x\rangle=DC_{4n}$.

2. We exhibit group morphisms $\varphi:DC_{4n}\rightarrow\Pi_{4n,n}$, $\psi:\Pi_{4n,n}\rightarrow DC_{4n}$, satisfying identities similar to 1.

Define $\varphi$ and $\psi$ on the generators:

[TABLE]

It is directly checked that $\varphi$ and $\psi$ can be extended to group morphisms from generators and identities $\psi\circ\varphi=id_{DC_{4n}}$, $\varphi\circ\psi=id_{\Pi_{4n,n}}$ are satisfied.

3. It follows that $y^{-1}by=b^{-1}$. Existence of a graph isomorphism follows from the uniqueness of Cayley graph construction as in the proof of Proposition 12. ∎

Example 18**.**

In Fig.2. continuous arrows mean left multiplication by $a^{2}$ and dotted arrows mean left multiplication by $x$.

\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{2}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{4}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{5}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{3}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{5}x\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{ax\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces} **

Fig.2. - the graph $\Gamma(DC_{12},(a^{2},x))$.

Remark 19**.**

By the Burnside Basis Theorem all minimal generating sequences of $DC_{2^{l}}$ must have length $2$. If $n$ is not a prime power, then $DC_{4n}$ has minimal generating sequences containing more than two elements. For $DC_{24}$ there are at least $6$ non-equivalent minimal generating sets containg three elements. For example, $DC_{24}$ has the following minimal generating sets: $\{a^{2},a^{3},x\}$ with orders $6,4,4$, $\{a^{3},a^{4},x\}$ with orders $4,3,4$, $\{a^{3},x,a^{2}x\}$ with orders $4,4,4$.

Acknowledgement

Computations were performed using the computational algebra system MAGMA, see Bosma et al. [1].