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Let G be a finite group and S be the set of Sylow p-subgroups of G for a prime p dividing the order of G. Assume that |S|>1. Let U and V be two disjoint non-empty subsets of S such that, $$\bigcup_{ …

I would like to give another solution; $G/G'=<xG'>$ for $x\in G$. It is easy to see that $G'=[x,G']$. Now the map $\phi:G'\to G'$ by $g\mapsto [g,x]$ is an homomorphism as $G'$ is abelain. More clea …

Theorem $1$(Burnside): A simple nonabelian finite group can not have a conjugacy classes with prime power elements. Theorem $2$: A group of order $p^nq^m$ is solvable. Theorem $1$ depends on charact …

Let $G$ be a $p$-group and $H$ be a subgroup of $G$. Set $V$ be a pretransfer map from $G$ to $H$. That is, $$V(g)=\prod_{t\in T} tg(t.g)^{-1}=\prod_{t\in T_0}tg^n_tt^{-1} $$ Can we say that the im …

Let $G$ be a nonabelian finite simple group all of whose Sylow subgroups of odd order are cyclic. If we further assume that its Sylow $2$-subgroup is dihedral, then due to Suzuki, we know that $G\cong …

Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd, Set $$a_1=g_1$$ $$a_2=g_1g_2$$ $$a_3=g_1g_2g_3$$ $$a_n=g_1g_2...g_n$$ I am looking for example that all $a_i$ are different from e …

Let $A$ be a frobenius complement for a group $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. A …

Let $G$ be a group and $\mathfrak{L}(G)$ be set of all subgroups of $G$. Clearly, $\mathfrak{L} (G)$ is a lattice. If we know that $\mathfrak{L} (G)$ is symmetric then what can be said about the gro …

Theorem:Let $P\in Syl_p(G)$ for a finite group $G$. Then $G$ has a normal $p-$ complement if and only if $P$ controls its own fusion. I wonder if similar argument is true for Hall subgroups (in gener …