On Odd Order Nilpotent Groups With Class 2

TL;DR

This paper investigates the structure of odd order nilpotent groups with class 2, establishing a lower bound on the number of non-isomorphic such groups based on the prime factorization of the exponent of their commutator subgroup.

Contribution

It provides a new lower bound on the count of non-isomorphic nilpotent groups of class 2 and odd order, related to the prime factorization of the commutator subgroup's exponent.

Findings

At least r_1 + r_2 + ... + r_s non-isomorphic groups exist for given parameters.

The number of groups depends on the prime factorization of the commutator subgroup's exponent.

The order of these groups matches that of the original group G.

Abstract

Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at least $r_1+r_2+... +r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$.