## Tokyo Journal of Mathematics

- Tokyo J. Math.
- Volume 42, Number 1 (2019), 113-120.

### Homology of the Complex of All Non-trivial Nilpotent Subgroups of a Finite Non-solvable Group

Nobuo IIYORI and Masato SAWABE

#### Abstract

Let $G$ be a finite non-solvable group. We study homology of the complex $\mathcal{N}(G)$ of all non-trivial nilpotent subgroups of $G$. The determination of $H_{n}(\mathcal{N}(G))$ is reduced to that of homology of its subcomplex $\mathcal{N}_{\pi_{1}}(G)$ consisting of all nilpotent $\pi_{1}$-subgroups, where $\pi_{1}$ is the connected component of the prime graph of $G$ containing 2. Furthermore, $\mathcal{N}_{\pi_{1}}(G)$ is connected if $G$ possesses no strongly embedded subgroups.

#### Article information

**Source**

Tokyo J. Math., Volume 42, Number 1 (2019), 113-120.

**Dates**

First available in Project Euclid: 18 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.tjm/1563436916

**Mathematical Reviews number (MathSciNet)**

MR3982052

**Zentralblatt MATH identifier**

07114903

**Subjects**

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups [See also 20F22]

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

#### Citation

IIYORI, Nobuo; SAWABE, Masato. Homology of the Complex of All Non-trivial Nilpotent Subgroups of a Finite Non-solvable Group. Tokyo J. Math. 42 (2019), no. 1, 113--120. https://projecteuclid.org/euclid.tjm/1563436916