Central elements of the Jennings basis and certain Morita invariants

TL;DR

This paper investigates Morita invariants related to the intersection of the center and socle layers of finite-dimensional algebras, with a focus on group algebras of p-groups and the Jennings basis.

Contribution

It proves that the intersection of the center and the nth socle is a Morita invariant and analyzes the structure of this intersection for group algebras of powerful p-groups.

Findings

The intersection of the center and the nth socle is a Morita invariant.

Certain Jennings basis elements are central in group algebras of powerful p-groups.

The socle of the group algebra is contained in the center for powerful p-groups.

Abstract

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the $n$th (right) socle $ZS^n(A) := Z(A) \cap \operatorname{Soc}^n(A)$ of a finite-dimensional algebra $A$ is a Morita invariant; This is a generalization of important Morita invariants --- the center $Z(A)$ and the Reynolds ideal $ZS^1(A)$. As an example, we also studied $ZS^n(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^n(FG)$. In fact, such elements form a basis of $ZS^n(FG)$ for every integer $1 \le n \le p$ if $G$ is powerful. As a corollary we have $\operatorname{Soc}^p(FG) \subseteq Z(FG)$ if $G$ is powerful.

Tables

Table 1. Table 1.1. What is known about Z ​ S n ​ ( F ​ G ) = Z ​ ( F ​ G ) ∩ Soc n ⁡ ( F ​ G ) 𝑍 superscript 𝑆 𝑛 𝐹 𝐺 𝑍 𝐹 𝐺 superscript Soc 𝑛 𝐹 𝐺 ZS^{n}(FG)=Z(FG)\cap\operatorname{Soc}^{n}\left({FG}\right) .

	dimension	basis
	(representation-theoretic)	(group-theoretic)
$Z\left(FG\right)$	$k\left(G\right)$	conjugacy class sums
$Z{S}^n\left(FG\right)$	unknown	unknown
$Z{S}^2\left(FG\right)$	$\ell \left(G\right)+\sum dim{\mathrm{Ext}}^1(S,S)$	unknown222Except finite $p$-groups; see Remark 4.9.
$Z{S}^1\left(FG\right)$	$\ell \left(G\right)$	$p$-regular section sums

Equations

$$Z{S}^n(A):=Z(A)\cap {\mathrm{Soc}}^n\left(A\right)$$

$$\dim Z{S}^2(FG)=\ell (G)+\sum \dim {\mathrm{Ext}}_A^1(S,S)$$

$$\ell \ell \left(A\right)=\min \{n\in \mathbb{N}|Z{S}^n(A)=Z(A)\}.$$

$$\sum _k{u}_{k}e{v}_k=1.$$

$$Z(A)$$

$$\phi\big{(}ZS^{n}(A)\big{)}\subseteq ZS^{n}(B)\quad\text{and}\quad\psi\big{(}ZS^{n}(B)\big{)}\subseteq ZS^{n}(A),$$

$$\dim Z{S}^3(F{\mathfrak{A}}_4)\ne \dim Z{S}^3(e_{0}F{\mathfrak{A}}_5)$$

$$\sum (a_{\lambda }z)\otimes {\epsilon }_{\lambda }=a(z\otimes 1)=0.$$

$$Z{S}^n(A{\otimes }_{F}E)=Z{S}^n(A){\otimes }_{F}E.$$

$$Z{S}^n(A{\otimes }_{F}E)$$

$$\displaystyle=\big{(}Z(A)\otimes_{F}Z(E)\big{)}\cap\operatorname{Soc}^{n}\left({A\otimes_{F}E}\right)$$

$$\displaystyle=\big{(}Z(A)\otimes_{F}E\big{)}\cap\big{(}\operatorname{Soc}^{n}\left({A}\right)\otimes_{F}E\big{)}$$

$$\displaystyle=\big{(}Z(A)\cap\operatorname{Soc}^{n}\left({A}\right)\big{)}\otimes_{F}E$$

$$=Z{S}^n(A){\otimes }_{F}E.$$

$$Z{S}^n(FG)=Z{S}^n({\mathbb{F}}_{p}G){\otimes }_{{\mathbb{F}}_p}F.$$

$$\dim Z{S}^n(A)\le \sum _i\dim e_{i}A{e}_i/e_i{\mathrm{Rad}}^n\left(A\right)e_i$$

$$D_i:=\{g\in G|g-1\in {\mathrm{Rad}}^i\left(FG\right)\}.$$

$${\mathrm{Rad}}^n\left(FG\right)=⨁F\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}1\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{m_{ij}}$$

$$\sum _{\begin{array}{c}1\le i<t\\ 1\le j\le r_i\end{array}}i{m}_{ij}\ge n.$$

$$\Bigg{\{}\,\sideset{}{{}^{\prime}}{\prod}_{\begin{subarray}{c}1\leq i<t\\ 1\leq j\leq r_{i}\end{subarray}}(g_{ij}-1)^{m_{ij}}\;\Bigg{|}\;0\leq{m_{ij}}<p\,\Bigg{\}}$$

$$\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}1\le i<s\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{m_{ij}}\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

$$Z{S}^n(FG)\supseteq ⨁F\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}1\le i<s\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{m_{ij}}\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

$$\sum _{\begin{array}{c}1\le i<s\\ 1\le j\le r_i\end{array}}i(p-1-m_{ij})<n.$$

$$\dim Z{S}^{n_s}(FG)\ge |G/D_s|$$

$$Z{S}^n(FG)=⨁F\text{sideset}{}^{\prime }{\prod }_{1\le j\le r_1}(g_{1j}-1{)}^{m_{1j}}\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}2\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

$$\sum _{1\le j\le r_1}(p-1-m_{1j})<n.$$

$$Z{S}^1(FG)$$

$$Z{S}^2(FG)$$

$$\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}=D_s^{+}.$$

$$\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

$$=\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}\sum _{m=0}^{p-1}(-1{)}^m(-1{)}^{p-1-m}{g}_{ij}^m$$

$$=\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}\sum _{m=0}^{p-1}{g}_{ij}^m=D_s^{+}.\text{qed}$$

$$FG\cdot D_s^{+}=⨁F\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}1\le i<s\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{m_{ij}}\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

$$\dim ⨁F\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}1\le i<s\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{m_{ij}}\text{sideset}{}^{\prime }{\prod }_{\begin{array}{c}s\le i<t\\ 1\le j\le r_i\end{array}}(g_{ij}-1{)}^{p-1}$$

Full text

As an example, we also studied $ZS^{n}(FG)$ for the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^{n}(FG)$. In fact, such elements form a basis of $ZS^{n}(FG)$ for every integer $1\leq n\leq p$ if $G$ is powerful. As a corollary we have $\operatorname{Soc}^{p}(FG)\subseteq Z(FG)$ if $G$ is powerful.

1. 1 Introduction
2. 2 Morita invariants
3. 3 Jennings theory
4. 4 Main theorems
5. 5 Proofs
6. 6 Examples

1. Introduction

From Morita theoretic viewpoint, computing Morita invariants is important to distinguish algebras that are not Morita equivalent. We prove that the intersection of the center and the $n$th (right) socle

[TABLE]

is Morita invariant for a finite-dimensional algebra $A$ (Theorem 2.3). This is a generalization of important Morita invariants — the center $Z(A)$ and the Reynolds ideal $ZS^{1}(A)$. The other way of generalization is known as the Külshammer ideals or the generalized Reynolds ideals for a finite-dimensional symmetric algebra; For more details we refer the reader to the survey by Zimmermann [18].

The centers and the Reynolds ideals are particularly interesting for finite group algebras. Let $G$ be a finite group and $F$ an algebraically closed field $F$ of positive characteristic $p$. First, the dimension of the center $Z(FG)$ equals the number of irreducible ordinary characters $k(G)$ and the dimension of the Reynolds ideal $ZS^{1}(FG)$ equals the number of irreducible modular characters $\ell(G)$. Next, the conjugacy class sums form a basis of the center and the $p$-regular section sums form a basis of the Reynolds ideal [8]. Moreover, Okuyama [13] proved

[TABLE]

where the sum is taken over a complete set of simple $FG$-modules. (In fact, he proved that for a block of a finite group algebra.) See also [7, Theorem 2.1] which is written in English. These are summarized in Table 1.1.

As $ZS^{n}(FG)$ is a generalization of these, we want to know what is the dimension and what a basis can be (Problem 2.8). One of manageable examples to compute socle series is the case for a finite $p$-group $G$. Jennings constructed group-theoretically a basis of $FG$ along the radical filtration (Theorem 3.5) and it follows that radical series coincides with socle series (Theorem 3.6). Such a basis is known as the Jennings basis (Definition 3.7). To study $ZS^{n}(FG)$ let us ask a question:

When is an element of the Jennings basis central?

We prove that certain elements of the Jennings basis are central and hence form a linearly independent set of $ZS^{n}(FG)$ (Theorem 4.1). In fact, such elements form a basis of $ZS^{n}(FG)$ for every integer $1\leq n\leq p$ if $G$ is powerful (Theorem 4.5). As a corollary we have $\operatorname{Soc}^{p}\left({FG}\right)\subseteq Z(FG)$ if $G$ is powerful (Corollary 4.6). Proofs are given in Section 5. Section 6 is devoted to illustrate our results by the first non-trivial examples.

2. Morita invariants

Héthelyi et al. [5, Corollary 5.3] proved that the Külshammer ideals are Morita invariants. (Later Zimmermann [17] proved that those are derived invariants as well.) Inspired by the proof we prove that $ZS^{n}(A)$, ideals of the center, are Morita invariants. The $n$th Jacobson radical of $A$ will be denoted by $\operatorname{Rad}^{n}\left({A}\right)$.

Lemma 2.1**.**

The Loewy length $\ell\ell\left(A\right)$ of a finite-dimensional algebra $A$ over a field can be expressed by $ZS^{n}(A)$ as

[TABLE]

Proof.

Clear. ∎

Lemma 2.2**.**

Let $A$ be a finite-dimensional algebra over a field and $e\in A$ a full idempotent. Then $\operatorname{Rad}^{n}\left({eAe}\right)=e\operatorname{Rad}^{n}\left({A}\right)e$ for every $n\in\mathbb{N}$.

Proof.

Since there is a canonical lattice isomorphism from ideals of $A$ to those of $eAe$ which preserves multiplication of ideals, the claim follows. (See [9, Theorem 21.11(2)].) ∎

Theorem 2.3**.**

Let $A$ and $B$ be Morita equivalent finite-dimensional algebras over a field. Then there is an algebra isomorphism $Z(A)\to Z(B)$ mapping $ZS^{n}(A)$ onto $ZS^{n}(B)$ for every $n\in\mathbb{N}$. In particular, $ZS^{n}(A)$ are Morita invariants.

Proof.

It suffices to prove the case for $A$ and its basic algebra $B:=eAe$ because Morita equivalent basic algebras are isomorphic. Since the basic idempotent $e\in A$ is full, there exist $u_{k},v_{k}\in A$ such that

[TABLE]

Then define algebra homomorphisms

[TABLE]

by $\phi(a)=eae$ for $a\in Z(A)$ and $\psi(b)=\sum_{k}u_{k}bv_{k}$ for $b\in Z(B)$. These are well-defined and mutually inverse. By (2.1) and Lemma 2.2 we have

[TABLE]

and the proof completes. ∎

Remark 2.4**.**

In general, $ZS^{n}(A)$ are not derived invariants. Let $F$ be an algebraically closed field of characteristic two, $\mathfrak{A}_{n}$ the alternating group of degree $n$, and $e_{0}$ the principal block idempotent of $F\mathfrak{A}_{5}$. Then $\ell\ell\left(F\mathfrak{A}_{4}\right)=3\neq 5=\ell\ell\left(e_{0}F\mathfrak{A}_{5}\right).$ In particular, by Lemma 2.1, we have

[TABLE]

although $F\mathfrak{A}_{4}$ and $e_{0}F\mathfrak{A}_{5}$ are derived equivalent. (cf. the Broué abelian defect conjecture.)

Now let us study how $ZS^{n}(A)$ behaves with field extensions.

Lemma 2.5**.**

Let $A$ be a finite-dimensional algebra over a field $F$ and suppose $A/\operatorname{Rad}\left({A}\right)$ is separable. Then for every field extension $E/F$ and $n\in\mathbb{N}$ we have the following.

1. (i)

$\operatorname{Rad}^{n}\left({A\otimes_{F}E}\right)=\operatorname{Rad}^{n}\left({A}\right)\otimes_{F}E$. 2. (ii)

$\operatorname{Soc}^{n}\left({A\otimes_{F}E}\right)=\operatorname{Soc}^{n}\left({A}\right)\otimes_{F}E$.

Proof.

(i): The proof is by induction on $n$ and [12, Lemma 2.5.1(ii)].

(ii): Let $a\in\operatorname{Soc}^{n}\left({A\otimes_{F}E}\right)$ and take an $F$-basis $\{\,\varepsilon_{\lambda}\;|\;\lambda\in\Lambda\,\}$ of $E$. Since $A\otimes_{F}E=\bigoplus A\otimes\varepsilon_{\lambda}$ we can express $a=\sum a_{\lambda}\otimes\varepsilon_{\lambda}$ for some $a_{\lambda}\in A$, which equals zero for all but finitely many $\lambda\in\Lambda$. For every $z\in\operatorname{Rad}^{n}\left({A}\right)$ we have $z\otimes 1\in\operatorname{Rad}^{n}\left({A\otimes_{F}E}\right)$ by (i). Hence

[TABLE]

By uniqueness we have $a_{\lambda}z=0$, which implies $a_{\lambda}\in\operatorname{Soc}^{n}\left({A}\right)$, for all $\lambda\in\Lambda$. We thus get $\operatorname{Soc}^{n}\left({A}\right)\otimes_{F}E\subseteq\operatorname{Soc}^{n}\left({A\otimes_{F}E}\right)$.

The proof for $\operatorname{Soc}^{n}\left({A}\right)\otimes_{F}E\subseteq\operatorname{Soc}^{n}\left({A\otimes_{F}E}\right)$ is easier. ∎

Proposition 2.6**.**

Let $A$ be a finite-dimensional algebra over a field $F$ and suppose $A/\operatorname{Rad}\left({A}\right)$ is separable. Then for every field extension $E/F$ and $n\in\mathbb{N}$ we have

[TABLE]

Proof.

[TABLE]

Corollary 2.7**.**

Let $F$ be a field of positive characteristic $p$ and $G$ a finite group. Then for every $n\in\mathbb{N}$ we have

[TABLE]

Proof.

Since $\mathbb{F}_{p}G/\operatorname{Rad}\left({\mathbb{F}_{p}G}\right)$ is separable [12, Lemma 3.1.28], the claim follows from Proposition 2.6. ∎

It would be interesting to study the following problem.

Problem 2.8** (See also Table 1.1).**

Let $G$ be a finite group and $p$ a prime divider of $\lvert{G}\rvert$. Construct bases and describe the dimensions of $ZS^{n}(\mathbb{F}_{p}G)$ for all $n\in\mathbb{N}$.

By Lemma 2.1, a satisfactory answer to this problem yields an answer to the Brauer Problem 15 [3] — a group-theoretic description of the Loewy length $\ell\ell\left(FG\right)$.

Remark 2.9**.**

Let $A$ be a block of a finite group algebra over a splitting field and $\{e_{i}\}$ a complete set of orthogonal primitive idempotents of $A$. Recently Otokita [14] proved an upper bound

[TABLE]

for every $n\in\mathbb{N}$.

3. Jennings theory

As we proved $ZS^{n}(A)$ are Morita invariaits, we want to determine these for special cases. Taking Problem 2.8 into account, we hereafter study the group algebra $FG$ of a finite $p$-group $G$ over a field $F$ of positive characteristic $p$. In this section, we collect some results of the Jennings theory for the reader’s convenience and to fix our notations.

Definition 3.1**.**

For $i\in\mathbb{N}$ we define the $i$th dimension subgroup (or Jennigs subgroup) of $G$ by

[TABLE]

Remark 3.2**.**

Although the dimension subgroups are defined ring-theoretically, these can be computed group-theoretically by Theorem 3.6(iv).

Lemma 3.3**.**

1. (i)

Dimension subgroups are a descending series of characteristic subgroups. 2. (ii)

Every successive quotient of the dimension subgroups is an elementary abelian $p$-group.

Proof.

See [6, 177–178]. ∎

We use the following notations throughout the paper.

Notation 3.4**.**

Let $D_{i}$ be the dimension subgroups of $G$. Set $t:=\min\{\,i\in\mathbb{N}\;|\;D_{i}=1\,\}$. Then we fix elements $g_{i1},\dotsc,g_{ir_{i}}\in D_{i}$ such that $\{\,g_{ij}D_{i+1}\;|\;1\leq j\leq r_{i}\,\}$ form a minimal generating set of $D_{i}/D_{i+1}$ for every integer $1\leq i<t$ satisfying $D_{i}>D_{i+1}$. We write $\sideset{}{{}^{\prime}}{\prod}$ for the product taken in lexicographic order with respect to indices.

Theorem 3.5** (Jennings).**

For every integer $n\geq 0$ we have

[TABLE]

where the direct sum is taken for all integers $0\leq{m_{ij}}<p$ satisfying

[TABLE]

Proof.

See [6, Theorem 3.2]. ∎

Theorem 3.6** (Jennings, Brauer).**

1. (i)

The Loewy length $\ell\ell\left(FG\right)$ of $FG$ equals $\displaystyle 1+(p-1)\sum_{1\leq i<t}ir_{i}$. 2. (ii)

$FG$* is rigid*:

$\operatorname{Soc}^{n}\left({FG}\right)=\operatorname{Rad}^{\ell\ell\left(FG\right)-n}\left({FG}\right)$* for every integer $0\leq n\leq\ell\ell\left(FG\right)$.* 3. (iii)

$\displaystyle\operatorname{Soc}^{n}\left({FG}\right)=\bigoplus F\sideset{}{{}^{\prime}}{\prod}_{\begin{subarray}{c}1\leq i<t\\ 1\leq j\leq r_{i}\end{subarray}}(g_{ij}-1)^{m_{ij}}$* for every integer $n\geq 0$ where the direct sum is taken for all integers $0\leq{m_{ij}}<p$ satisfying $\displaystyle\sum_{\begin{subarray}{c}1\leq i<t\\ 1\leq j\leq r_{i}\end{subarray}}i(p-1-{m_{ij}})<n$.* 4. (iv)

$D_{1}=G$* and $D_{i}=(D_{\lceil i/p\rceil})^{p}[D_{i-1},G]$ for every integer $i>1$.*

Proof.

1. (i)

See [6, Theorem 3.7]. 2. (ii)

See [1, Proof of Corollary 3.14.7]. 3. (iii)

The proof follows from part (ii) and Theorem 3.5. 4. (iv)

See [6, Theorem 5.5]. ∎

Definition 3.7**.**

The basis

[TABLE]

of $FG$ is said to be the Jennings basis.

4. Main theorems

Recall Notation 3.4 in the following.

Theorem 4.1**.**

Suppose $s\in\mathbb{N}$ satisfies $D_{s}\geq[G,G]$. Then an element of the Jennings basis of the form

[TABLE]

are central for every integers $0\leq{m_{ij}}<p$. In particular, for every $n\in\mathbb{N}$, we have

[TABLE]

where the direct sum is taken for all integers $0\leq{m_{ij}}<p$ satisfying

[TABLE]

Remark 4.2**.**

Note $D_{2}\geq[G,G]$.

Remark 4.3**.**

From (4.1), we have a lower bound

[TABLE]

where $\displaystyle n_{s}:=1+(p-1)\sum_{1\leq i<s}ir_{i}$.

See Section 6 for concrete examples. We can show that equality holds in (4.1) for the following class of $p$-groups.

Definition 4.4** (Lubotzky-Mann [10]).**

A finite $p$-group $G$ is said to be powerful if $G^{p}\geq[G,G]$ and $p>2$, or $G^{4}\geq[G,G]$ and $p=2$.

Theorem 4.5**.**

If $G$ is powerful then, for every integer $1\leq n\leq p$, we have

[TABLE]

where the direct sum is taken for all integers $0\leq m_{1j}<p$ satisfying

[TABLE]

Corollary 4.6**.**

If $G$ is powerful then $\operatorname{Soc}^{p}\left({FG}\right)\subseteq Z(FG)$.

Remark 4.7**.**

Even if $G$ is powerful, the assertion $\operatorname{Soc}^{p+1}\left({FG}\right)\subseteq Z(FG)$ is false. See Subsection 6.2.

Remark 4.8**.**

Let $A$ be a finite-dimensional symmetric algebra over a field. In general, it is known that $\operatorname{Soc}^{2}\left({A}\right)\subseteq Z(A)$ if $A$ is split-local. This can be traced back to Müller [11, Proof of Lemma 2]. (For a simple proof see, for example, [2, Lemma 2.2].)

Remark 4.9**.**

Note that $ZS^{n}(FG)$ can be described explicitly for $n=1,2$.

[TABLE]

This is due to the Jennings theory and Remark 4.8.

Remark 4.10**.**

It is, of course, not true that $ZS^{n}(FG)$ is always spanned by a subset of the Jennings basis; Any 2-group of maximal class of order 16 yields a minimal counterexample [4, 15].

5. Proofs

5.1. Proofs of lemmas

A certain ideal associated to a normal subgroup plays an important role in the proof of the main theorems. We prove its properties in the following.

Lemma 5.1**.**

Let $G$ be a finite group, $N\unlhd G$, and $F$ a field. Let $N^{+}$ denote the sum of all elements of $N$ in $FG$. If $N\geq[G,G]$ then $FG\cdot N^{+}\subseteq Z(FG)$.

Proof.

Clear. ∎

Remark 5.2**.**

The $N=[G,G]$ case of Lemma 5.1 can be found in [16, Lemma 5].

Lemma 5.3**.**

[TABLE]

Proof.

[TABLE]

Lemma 5.4**.**

[TABLE]

where the direct sum is taken for all integers $0\leq{m_{ij}}<p$.

Proof.

First, note that the dimensions are equal because

[TABLE]

By Lemma 5.3, we have

[TABLE]

and the proof is complete. ∎

Lemma 5.5**.**

If $G$ is powerful then $D_{2}=D_{p}$.

Proof.

The proof follows from Theorem 3.6(iv). ∎

5.2. Proof of main theorems

Proof of Theorem 4.1.

The first part follows from

[TABLE]

The second part follows from (5.1) and Theorem 3.6(iii). ∎

Proof of Theorem 4.5.

By Theorem 4.1, it remains to prove

[TABLE]

We claim that

[TABLE]

where the direct sum is taken for all integers $0\leq m_{1j}<p$ satisfying $\displaystyle\sum_{1\leq j\leq r_{1}}(p-1-m_{1j})<n$. Let

[TABLE]

be an element of the Jennings basis lying in $\operatorname{Soc}^{n}\left({FG}\right)$. Then we have the following.

[TABLE]

Hence we can express

[TABLE]

as we claimed. ∎

Proof of Corollary 4.6.

The proof follows from the claim (5.2) and Theorem 4.1. ∎

6. Examples

In this section, we illustrate our results by the first non-trivial examples: group algebras of extra-special $p$-groups of order $p^{3}$ for odd prime $p$.

6.1. Extra-special $p$-group $p_{+}^{1+2}$

Let $G$ be an extra-special $p$-group of order $p^{3}$ and exponent $p$ defined by

[TABLE]

and set $x:=a-1$, $y:=b-1$, and $z:=c-1$. Then $G$ is not powerful and the dimension subgroups of $G$ are

[TABLE]

We can show the following.

[TABLE]

6.2. Extra-special $p$-group $p_{-}^{1+2}$

Let $G$ be an extra-special $p$-group of order $p^{3}$ and exponent $p^{2}$ defined by

[TABLE]

and set $x:=a-1$, $y:=b-1$, and $z:=c-1$ where $c=b^{p}$. Then $G$ is powerful and the dimension subgroups of $G$ are

[TABLE]

We can show the following.

[TABLE]

In particular, we have

[TABLE]

as expected. Note $x^{p-1}y^{p-1}z^{p-2}\in\operatorname{Soc}^{p+1}\left({FG}\right)\setminus Z(FG)$.

6.3. Remark

In the above examples, there are two series of elements of $ZS^{n}(FG)$: Of the form $x^{i}y^{j}z^{p-1}$ and $z^{k}$. Our main theorem states that an element of the Jennings basis like $x^{i}y^{j}z^{p-1}$ is always central for all $p$-groups.

Acknowledgments

The author wishes to acknowledge Professor Shigeo Koshitani and Dr. Yoshihiro Otokita for helpful discussions.