Let $G$ be a finite group and $Sub(G)$ denote the number of subgroups of $G$ including the trivial subgroup and $G$ itself. I believe the following is true:$Sub(H \times K)\leq Sub(H \rtimes K)$ for ...

Fix a prime $p$, and two finite $p$-groups $G$ and $H$. Then we can try to find a group $K$ of minimal order such that there exists a surjective homomorphism $\alpha\colon K\to G$ and another ...

Background:Let $G$ be a finite group. Fix a prime $p$. We say that $H\subseteq G$ is strongly $p$-embedded if:$p$ divides $|H|$;For every $x\in G-H$, $p$ does not divide $|H\cap H^x|$.Some facts ...

Let $G$ be a finite group and $\phi : G \rightarrow G$ be a group automorphism such that for more than $\frac{3}{4}$ of elements $g \in G$ we have$$\phi(g) = g^{-1}$$Prove that for all $g \in G$$$\...

Let $G$ be a finite group and $p$ be any prime number dividing $|G|$. Let $S_p(G)$ be the set of all $p$-singular elements of $G$ i.e. elements whose orders are divisible by $p$.Conjecture: The ...

Let $p$ be a prime, $p\neq 2$. Let $Q$ be a cyclic $2$-group and $P$ an elementary abelian $p$-group of rank $n$. Suppose that $Q$ acts faithfully on $P$ (so $Q$ is isomorphic to a subgroup of $GL(n,...

Is there any classification of finite simple groups of order $4m$, where $m$ is an odd square free integer?Some examples are of order $60, 660, 1092, 12180, 102660$.

A set of $k$ generators is called irredundant if it is minimal by inclusion, i.e. any $k-1$ of them generate a proper subgroup. I wonder if there's a sampling procedure for such families of generators ...

Consider the free poly-nilpotent groups $F⁄\gamma_{i_k}( …\gamma_{i_2} (\gamma_{i_1 }(F ))…)$. Are they residually finite?My attempts to present this group as an extension of more understandable ...

Let $G$ be a group, $R$ a ring, and $0\neq\chi:G\to\mathbb{R}$. Define the Novikov completion$$\widehat{R G}^{\chi}=\Bigl\{\sum_{g\in G} r_g g \ \Big|\ \#\{g\mid r_g\neq 0,\ \chi(g)\le t\}<\...

The simplest possible complex reflection group is the cyclic group of order $n$ acting on $\mathbb C$ by rotations. This is also the orientation-preserving subgroup of the dihedral group of order $2n$,...

Let $\Gamma$ be a torsion free Zariski-dense discrete subgroup of $\operatorname{SL}_3(\mathbb{R})$. Then one can show that the cohomological dimension of $\Gamma$ is less than or equal to 5. The ...

Let $G$ be a finite perfect group with non-trivial center. I am trying to show that a Sylow $2$-subgroup of $G$ is non-abelian.I think one needs to use transfer homomorphisms here. However I can not ...

For an infinite set S, I believe that all automorphisms of Sym(S) are inner. I would like a reference for this.

Let $G$ be a finite group, and fix a prime $p$ which divides $|G|$.The ordinary (complex) characters $\text{Irr}(G)$ can be partitioned into what are called $p$-blocks, and to each block $B$ can be ...