# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

1,489 questions
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### Zero divisors in complex group algebras of residually finite groups

Conjecture. There exists a function $f:\mathbb{N} \rightarrow \mathbb{N}$ such that if $\alpha$ and $\beta$ are non-zero elements of the complex group algebra $\mathbb{C}[G]$ of a finite group $G$ ...
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### Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$

When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for ...
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### Number of elementary abelian subgroups of extraspecial $2$-groups

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following ...
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### lifting of idempotents in group ring

Let $G$ be a finite group, and let $\pi:G\to Q$ be a surjective group homomophism. The map $\pi:G\to Q$ does not necessarily split, but we can always find a set theoretical splitting $s:Q\to G$. In ...
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### Groups whose poset of direct factors are lattices

Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...
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### Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups

I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table. More than that, I would like to know the irreducible ...
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### Given any group G, how can we construct another group H such that G is isomorphic to the commutator subgroup H' of H?

First of all, it's not true that any group can be realized as the commutator subgroup of some group. So, if we assume that there is atleast one group H with H' isomorphic to G, how to construct all ...
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### Normalizers of abelian Sylows in simple groups

Suppose $G$ is a (nonabelian) finite simple group and $p$ is a prime such that the $p$-Sylow in $G$ is abelian. What can be said about its normalizer? I'm particularly interested in lower bounds on ...
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### Schreier conjecture — without a simple proof? and sporadic simple groups

The Schreier conjecture asserts that $\mathrm{Out}(G)$ is always a solvable group when $G$ is a finite simple group. This result is known to be true as a corollary of the classification of finite ...
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### Certain $p$-group with cyclic center
Let $G$ be a finite $p$-group of derived length $d$, which is not a Dedekind group. (i.e., possesses at least one non-normal subgroup). Let $G^{(d-1)}$ be the unique normal subgroup of $G$ of order \$...