On the Schur multiplier of $p$-groups with abelianization $s$-elementary abelian

Abstract: Let $p$ be an odd prime. We describe a method to compute the Schur multiplier of finite $p$-groups $G$ of nilpotency class $2$ such that $G/[G,G]$ is isomorphic to direct product of copies of $\mathbb{Z}_{p<sup>s}$ for $s \in \mathbb{N}$, generalizing a method of Blackburn and Evens, who treated the case $s=1$. As an application, we investigate which abelian $p$-groups can occur as the Schur multiplier of a non-abelian $p$-group. We further introduce the notions of $s$-special $p$-groups of rank $k$ generalizing the notion of special $p$-groups of rank $k$. We study the structural properties, compute the Schur multipliers of $s$-special $p$-groups of rank $1$.

• The paper establishes an explicit formula for |M(G)| by linking tensor product quotients and exterior product components.
• It extends Blackburn and Evens’ methods to p-groups with s-elementary abelianization, resolving a Kourovka notebook problem on Schur multiplier realizability.
• The study introduces s-special p-groups with constructive generators and numerical bounds, enhancing our approach to group extension classifications.

On the Schur Multiplier of $p$-Groups with Abelianization $s$-Elementary Abelian

Introduction

The computation of the Schur multiplier $M(G) = H^2(G, \mathbb{C}^\times)$ for finite $p$-groups is a central problem in group cohomology and representation theory, with ramifications in the classification of group extensions and invariants of $p$-groups. This work addresses the Schur multiplier structure for nilpotency class $2$ $p$-groups whose abelianization $G/G'$ is isomorphic to an $s$-elementary abelian $p$-group, i.e., $s$0. The authors extend methods due to Blackburn and Evens, who considered the case $s$1, adapting these computations to arbitrary $s$2. The paper also investigates which abelian $s$3-groups occur as Schur multipliers of non-abelian $s$4-groups, addressing a problem from the Kourovka notebook, and develops the theory of $s$5-special $s$6-groups, a generalization of classical special $s$7-groups.

Methodology: Computing the Schur Multiplier

Let $s$8 be a finite $s$9-group ($M(G) = H^2(G, \mathbb{C}^\times)$0 odd) of class $M(G) = H^2(G, \mathbb{C}^\times)$1, with $M(G) = H^2(G, \mathbb{C}^\times)$2 an $M(G) = H^2(G, \mathbb{C}^\times)$3-elementary abelian $M(G) = H^2(G, \mathbb{C}^\times)$4-group and $M(G) = H^2(G, \mathbb{C}^\times)$5. The structure of $M(G) = H^2(G, \mathbb{C}^\times)$6 is described via the second cohomology group and exterior products involving $M(G) = H^2(G, \mathbb{C}^\times)$7 and $M(G) = H^2(G, \mathbb{C}^\times)$8. The authors define the subgroups $M(G) = H^2(G, \mathbb{C}^\times)$9 and $p$0 of $p$1, arising from specific commutator relations and $p$2-th powers respectively, and consider the quotient $p$3.

A key morphism $p$4 is induced by the commutator pairing. The main technical result, Theorem 1, establishes that there exists an abelian group $p$5 with subgroup $p$6 and $p$7, such that the Schur multiplier is isomorphic to an explicit subgroup $p$8. The construction uses pullback extensions and central extensions, synthesizing homological algebra and explicit group theory techniques. The precise counting result is given as

$p$9

The authors employ presentations, Hall-Petrescu commutator relations, and module-theoretic arguments to substantiate these claims.

Application to the Kourovka Problem

The authors utilize the developed machinery to address which abelian $p$0-groups can be realized as the Schur multiplier of a non-abelian $p$1-group, a question listed as Problem 15.30 in the Kourovka notebook. The main results demonstrate that for any $p$2, the following abelian $p$3-groups arise as Schur multipliers:

• $p$4 for $p$5,
• $p$6 (with explicit constraints on $p$7, $p$8, $p$9, and $2$0),
• $2$1 for any $2$2 and parameters with $2$3, provided $2$4, $2$5.

In particular, groups of the form $2$6 for arbitrary non-negative integers $2$7 can be achieved as Schur multipliers.

The argument constructs $2$8-groups with prescribed abelianization and commutator structure to engineer the required Schur multiplier, using the theory developed for class $2$9 groups with $p$0-elementary abelian abelianization and $p$1-elementary abelian commutator subgroup.

$p$2-Special and $p$3-Extraspecial $p$4-Groups

The concept of $p$5-special $p$6-groups of rank $p$7, defined by $p$8 and both $p$9, $G/G'$0 isomorphic to $G/G'$1, generalizes classical special $G/G'$2-groups ($G/G'$3). The paper obtains explicit structural results for $G/G'$4-extraspecial $G/G'$5-groups (the $G/G'$6 case), showing that every such group is a central product of groups of order $G/G'$7 amalgamating the center, analogous to the central product decomposition for classical extraspecial groups. Specifically, if $G/G'$8 is an $G/G'$9-extraspecial $s$0-group and $s$1 with $s$2, then

• $s$3 is the central product of $s$4 many $s$5-extraspecial subgroups of order $s$6,
• $s$7 for $s$8.

Furthermore, for $s$9, such groups are shown to be unicentral and hence not capable.

Numerical Highlights and Explicit Generators

• Explicit Orders: For an $p$0-extraspecial group $p$1 of order $p$2, $p$3.
• Sharp Realizability: The constrained abelian group realizations for $p$4 are tight; the bounds on the number of $p$5 summands are achieved except possibly for a finite set of exceptions.
• Construction of Generators: The generators of the kernel of $p$6 are explicitly described in terms of commutator structure, providing not only existential but constructive information.

Theoretical and Practical Implications

The extension of the computation of Schur multipliers from elementary abelian to homocyclic abelianizations increases the landscape of possible invariants for $p$7-groups of class $p$8 and extends group extension classification. The explicit construction of groups $p$9 for prescribed $s$00 informs the inverse direction of the Schur multiplier problem, which is nontrivial even for $s$01-groups.

On a theoretical level, the techniques advance the process of determining second cohomology for finite $s$02-groups, connect with the structure theory of central products, and inform the study of group capability. Practically, the results enable the generation of classes of $s$03-groups with given extension and representation theoretic properties, which may be relevant for algorithmic group theory and computational cohomology.

Potential Future Directions

Further exploration could include:

• Extension of the methods to $s$04-groups of higher nilpotency class.
• Classification of Schur multipliers for more general abelianizations or commutator structures.
• Investigation of minimal presentations for $s$05-groups realizing a given abelian group as Schur multiplier.
• Analysis of impact on the classification of $s$06-group extensions and connection to automorphism group structure.

Conclusion

This paper generalizes and unifies approaches for computing the Schur multiplier of a broad class of nilpotency class $s$07 $s$08-groups with abelianization of homocyclic type, delivers explicit structural and numerical results, and solves realization problems for abelian $s$09-groups as Schur multipliers. The developed theory of $s$10-special $s$11-groups further enriches the taxonomy of $s$12-groups and clarifies the interplay between commutator structure and cohomology. The results provide new tools and perspectives for the study of group extensions, cohomological invariants, and the realization problem in the context of finite group theory.

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For reference and further details, see "On the Schur multiplier of $s$13-groups with abelianization $s$14-elementary abelian" (2605.00810).