Articles of group theory

Splitting of conjugacy class in alternating group

Browsing the web I came across this: The conjugacy class of an element : splits if the cycle decomposition of comprises cycles of distinct odd length. Note that the fixed points are here treated as cycles of length , so it cannot have more than one fixed point; and does not split if the cycle […]

Dedekind modular law

Dedekind modular law. If $A,B,C$ are subgroups of a group $G$ with $A \subseteq B$ then $A(B \cap C) = B \cap AC$. Below is what I want to prove. Let K be a finite group with $K = LH$, where $L,H$ are subgroups of $K$ with relatively prime orders. If $U$ is a maximal […]

Structure Theorem for abelian torsion groups that are not finitely generated

I know about the structure theorem for finitely generated abelian groups. I’m wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In particular, I’m interested in torsion groups. Maybe having a finite exponent helps?

Proving that $G/N$ is an abelian group

Let $G$ be the group of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$ where $ad \neq 0$ under matrix multiplication. Let $N=\left\{A \in G \; \colon \; A = \begin{pmatrix}1 & b \\ 0 & 1\end{pmatrix} \right\}$ be a subset of the group $G$. Prove that […]

Solvability of a group with order $p^n$

If $G$ is a group whose order is $p^n$($p$ is prime), then $G$ is solvable. How am I going to show this? Any help is appreciated. Thank you.

How can I prove that every group of $N = 255$ elements is commutative?

There was previous task was same but with $N = 185$. And I prove it by showing that number of Sylow subgroups is 1 for every prime $p\mid N$. But there I have some options $N_5 \in \{1, 51\}$, $N_17 = 1$, $N_3 \in \{1, 85\}$. I’ve tried to get contradiction from $N_5 = 51$ […]

Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.

Let $G$ a group of order $6$. Prove that: i) $G$ contains 1 or 3 elements of order 2. ii) $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$. I haven´t covered Sylow groups and normal groups. This is an exercise from the chapter about group actions. I have covered Lagrange and cosets.

Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative

This question already has an answer here: If $G/Z(G)$ is cyclic, then $G$ is abelian 2 answers

Let G be an abelian group, and let a∈G. For n≥1,let G := {x∈G:x^n =a}. Show that G is either empty or equal to αG := {αg : g ∈ G}…

We were given questions to study for our exam coming up. We have not covered much of this topic, so any help would be greatly appreciated! Let $G$ be an abelian group, and let $a\in G$. For $n≥1$, let $G[n:a] := \{x\in G:x^n =a\}$. (a) Show that $G[n: a]$ is either empty or equal to […]

Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.

(i) Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group. I understand isomorphism to be a bijective homomorphism but I’m unsure how one would go about proving this. $S_4/V$ has order 6 and I think that will be of use but that’s as far as I can go. (ii) […]