## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 47, Number 1-2 (2003), 49-62.

### On capable $p$-groups of nilpotency class two

Michael R. Bacon and Luise-Charlotte Kappe

#### Abstract

A group is called capable if it is a central factor group. Let ${\mathcal{P}}$ denote the class of finite $p$-groups of odd order and nilpotency class 2. In this paper we determine the capable 2-generator groups in ${\mathcal{P}}$. Using the explicit knowledge of the nonabelian tensor square of 2-generator groups in ${\mathcal {P}}$, we first determine the epicenter of these groups and then identify those with trivial epicenter, making use of the fact that a group has trivial epicenter if and only if it is capable. A capable group in ${\mathcal{P}}$ has the two generators of highest order in a minimal generating set of equal order. However, this condition is not sufficient for capability in ${\mathcal{P}}$. Furthermore, various homological functors, among them the exterior square, the symmetric square and the Schur multiplier, are determined for the 2-generator groups in ${\mathcal{P}}$.

#### Article information

**Source**

Illinois J. Math., Volume 47, Number 1-2 (2003), 49-62.

**Dates**

First available in Project Euclid: 17 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258488137

**Digital Object Identifier**

doi:10.1215/ijm/1258488137

**Mathematical Reviews number (MathSciNet)**

MR2031305

**Zentralblatt MATH identifier**

1030.20009

**Subjects**

Primary: 20D15: Nilpotent groups, $p$-groups

Secondary: 19C09: Central extensions and Schur multipliers

#### Citation

Bacon, Michael R.; Kappe, Luise-Charlotte. On capable $p$-groups of nilpotency class two. Illinois J. Math. 47 (2003), no. 1-2, 49--62. doi:10.1215/ijm/1258488137. https://projecteuclid.org/euclid.ijm/1258488137