[Paper Review] A characterization of finite $p$-groups by their Schur multiplier

Sumana Hatui|arXiv (Cornell University)|Aug 22, 2016

Finite Group Theory Research9 references3 citations

TL;DR

This paper classifies non-abelian finite $p$-groups $G$ of order $p^n$ for which the Schur multiplier $M(G)$ satisfies $t(G) = \log_p(|G|) + 1$, where $t(G)$ is the corank of $G$. Using group-theoretic techniques, including the Schur multiplier formula, isoclinism classification, and cohomological bounds, the authors fully characterize such groups for odd $p$ and $p=2$, identifying 12 isomorphism classes for odd $p$ and 6 for $p=2$, with explicit presentations provided.

ABSTRACT

Let $G$ be a finite $p$-group of order $p^n$ and $M(G)$ be its Schur multiplier. It is well known result by Green that $|M(G)|= p^{\frac{1}{2}n(n-1)-t(G)}$ for some $t(G) \geq 0$. In this article we classify non-abelian $p$-groups $G$ of order $p^n$ for $t(G)=\log_p(|G|)+1$.

Motivation & Objective

• To classify non-abelian finite $p$-groups $G$ of order $p^n$ for which the Schur multiplier satisfies $t(G) = \log_p(|G|) + 1$, where $t(G)$ is the corank defined by $|M(G)| = p^{\frac{1}{2}n(n-1) - t(G)}$.
• To extend prior classifications of $p$-groups by their Schur multiplier, particularly for higher values of $t(G)$, beyond the known cases $t(G) = 0,1,2,3$.
• To determine the complete list of isomorphism types of such $p$-groups using structural group theory, isoclinism invariants, and cohomological bounds on the multiplier.
• To resolve the case $t(G) = \log_p(|G|) + 1$, which corresponds to $s(G) = 3$ in Peyman Niroomand's generalized corank framework, and to establish the full structure of these groups.

Proposed method

• Use of the Schur multiplier formula $|M(G)| = p^{\frac{1}{2}n(n-1) - t(G)}$ to define the corank $t(G)$, and relate it to the generalized corank $s(G)$ via $|M(G)| = p^{\frac{1}{2}(n-1)(n-2)+1-s(G)}$.
• Application of Theorem 2.1 (from [17]) to bound $|M(G)|$ via central quotients, relating $|M(G)|$ to $|M(G/K)|$, $|G/G' \otimes K|$, and $|G' \cap K|$.
• Employment of James' isoclinism classification of $p$-groups of order $p^n$ for $n \leq 6$, denoted $\Phi_k$, to enumerate candidate groups.
• Use of Theorem 2.3 and Theorem 2.4 to bound the size of the Schur multiplier via images of the bilinear maps $\psi_2$ and $\psi_3$, analyzing the rank of $X$ and $X_1$ in the tensor product $G' \otimes G/G'$.
• Systematic case analysis based on the order of $G'$ and $Z(G)$, distinguishing cases by $|G'| = p$, $p^2$, or $p^3$, and using central quotient arguments to reduce to known multiplier bounds.
• Computational verification using HAP in GAP for $p=2$ to eliminate non-existent groups, and structural checks via presentations and exponent conditions.

Experimental results

Research questions

• RQ1 Which non-abelian $p$-groups $G$ of order $p^n$ satisfy $t(G) = \log_p(|G|) + 1$?
• RQ2 What is the complete list of isomorphism types of such $p$-groups for odd $p$ and for $p=2$?
• RQ3 How do the structures of $G$, $G'$, $Z(G)$, and $G/G'$ constrain the Schur multiplier size in this case?
• RQ4 What role does the generalized corank $s(G) = 3$ play in determining the isomorphism type of $G$?
• RQ5 Are there any $p$-groups of exponent $p^2$ or $p^3$ satisfying this condition, and if so, how are they characterized?

Key findings

• For odd $p$, the non-abelian $p$-groups with $t(G) = \log_p(|G|) + 1$ are isomorphic to one of 12 specific groups: $\Phi_2(22)$, $\Phi_3(211)a$, $\Phi_3(211)b_r$, $\Phi_2(2111)c$, $\Phi_2(2111)d$, $\Phi_3(1^5)$, $\Phi_7(1^5)$, $\Phi_{11}(1^6)$, $\Phi_{12}(1^6)$, $\Phi_{13}(1^6)$, $\Phi_{15}(1^6)$, and $(\mathbb{Z}_p^{(4)} \rtimes \mathbb{Z}_p) \times \mathbb{Z}_p^{(2)}$.
• For $p=2$, the groups are isomorphic to one of 6 groups: $\mathbb{Z}_2^{(4)} \rtimes \mathbb{Z}_2$, $\mathbb{Z}_2 \times ((\mathbb{Z}_4 \times \mathbb{Z}_2) \rtimes \mathbb{Z}_2)$, $\mathbb{Z}_4 \rtimes \mathbb{Z}_4$, $D_{16}$, and two others from the isoclinism list.
• The Schur multiplier of all such groups has order $p^8$, as $|M(G)| = p^{\frac{1}{2}n(n-3)-1}$ with $n=6$ for $p$ odd and $n=5$ for $p=2$, yielding $|M(G)| = p^8$ in both cases.
• Groups with $|G'| = p^3$ and $t(G) = \log_p(|G|)+1$ must have $Z(G) \subseteq G'$ and $|Z(G)| \geq p^2$, and are isomorphic to $\Phi_{11}(1^6)$ for odd $p$.
• For $p=2$, no such group exists with $|G'| = p^3$, and the case $|G'| = p^2$ leads to $|M(G)| < p^8$, so only groups with $|G'| = p$ or $p^2$ are possible.
• The group $\Phi_{11}(1^6)$ is the only group of exponent $p$ with $|G'| = p^3$, $|Z(G)| \geq p^2$, and $|M(G)| = p^8$, and it is the unique solution in this class.