Articles of group theory

Kernel of action of G on set of cosets of H in G is contained in H

Let $G$ be a group and $H$ a subgroup of G. Also, let $X$ be the set of left cosets, $xH$, of H in G. Define an action of $G$ on $X$ by $g \cdot xH = gxH$ for $g,x \in G$. I have shown that the kernel, $K$, where $K=\bigcap_{xH \in X}xHx^{-1}$ and $Stab_{G}(xH) […]

Free Product of two finite groups

The question entails that I should choose two finite groups, then construct a ‘biregular’ tree, and show that the action of the free product of the two finite groups on the biregular tree will have a fundamental domain that consists of a single edge and two vertices. What I have so far is the two […]

centralizer of transvection

let $T=T_{a,u}(v)$ be transvection, I want to find centralizer of $T_{a,u}(v)$ in $GL(V)$. What is $C _{GL(V)}(T)$? $T_{a,u}(v)=v+u(v)a$ where $a$ is vector in vector space $V$ on field $F$ and $u$ is linear functional on $V$.It is not hard to see $T_{a,u}(v)\in SL(V)$ I have not any idea how to deal with it.

if $Q$ and $P$ are distinct $p$-Sylow subgroups then $Q\not\subseteq N_G(P)$.

I have been told to use the following to prove another claim, but I would like to prove this anyway for myself. However I can’t tell why it’s true. I think it’s true, but can’t see why! Here it is: If $P$ and $Q$ are distinct $p$-Sylow subgroups in a group $G$ then $Q\not\subseteq N_G(P)$. […]

$|F:H|=|F/N:H/N|$ for nonnormal $H$

This should be very elementary, but the solution evades me. Suppose we have a tower of groups: $$ N \le H \le F $$ with N normal of infinite index in F, and H nonnormal of finite index in F. How one can show that $$|F:H|=|F/N:H/N|$$ where $|A:B|$ denotes the index of subgroup $B$ in […]

If $G$ is a finite group, $H\vartriangleleft G$, $G/H$ is finite $p$-group and $H\subseteq Z\left(G\right)$, show that $$ is $p$-group

If $G$ is a finite group, $H\vartriangleleft G$, $G/H$ is finite $p$-group and $H\subseteq Z\left(G\right)$, show that $[G,G]$ is $p$-group.

Show that the quotient group $T/N$ is abelian

let $T= \begin{bmatrix}a&b\\0&d\end{bmatrix}, a, b, d \in \mathbb{R}, ad \neq 0$ And let $N = \begin{bmatrix}1&b\\0&1\end{bmatrix}, b\in \mathbb{R}$ I showed that N is a normal subgroup of T Now I have to show that the quptient group $\dfrac{T}{N}$ is an abelian group (I am not sure what that quotient group represents). I would need a […]

Mapping between subgroups by an isomorphism.

The original question that I wrote was: A subgroup characteristic of the whole group I was wishing that there is an argument that is simple enough to see that “clearness” since the book that I am reading (Isaacs’ Algebra) recommended to accept a isomorphism as a map carrying any group theoretic property so that it […]

understand quotient group

i am trying to understand what does mean quotient terminology in group theory by as simple way as possible,also quotient group i want to know something about it,using internet i read that ” In mathematics, specifically group theory, a quotient group (or factor group) is a group obtained by aggregating similar elements of a larger […]

If $gh = hg, \ \ \gcd(|g|, |h|) = 1$, then $|gh| = |g||h|$($|a|$ is the order of element $a$ in a group $G$)

Let $G$ be a group and $g,h \in G$. I need to prove that if $g$ and $h$ commute and their orders are coprime, then $|gh| = |g||h|$, that is, the order of their product is the multiple of their orders. Since $gh = hg$, then $(gh)^{ lcm(|g|, |h|)} = e$, so, $|gh|$ divides $lcm(|g|, […]