Articles of group theory

Checking understanding on proving uniqueness of identity and inverse elements of a group.

Sorry for such a trivial question, but just wanted to check my understanding. When proving a statement, for example, that the inverse of a group element is unique (in elementary group theory) one starts by supposing that there exists two inverses $h$ and $k$ for a given element $g \in G$, where $G$ is some […]

Proving the Thompson Transfer Lemma

Let $G$ be a finite group of even order $n=2^kr$, $T$ a Sylow-$2$ subgroup of $G$, and $M$ an index $2$ subgroup of $T$. I want to show that if $G$ has no subgroup of index $2$, then every element $x$ of order $2$ is conjugate to an element of $M$. Using the transfer homomorphism, […]

Abelianization of general linear group?

I am asking purely out of interest: What the abelianization of general linear group $GL(n,\mathbb{R})$?

Is there a homomorphism from a full product of finite cyclic groups onto $\mathbb Z$?

Trying to answer this question, I encountered the following question, the answer to which should be known but it is hard to Google, so I did not find it. Let $G=\prod_{n\in\mathbb N}\mathbb Z_n$ be a full product of finite cyclic groups $\mathbb Z_n=\mathbb Z/n\mathbb Z$. Exists there a homomorphism $f:G\to\mathbb Z$ such that $f((1,1,\dots))=1$? According […]

Non abelian group with normal subgroup

Normal Subgroups are subgroups where all left cosets are right cosets. For abelian groups all subgroups are normal. I want to discuss about a non-abelian group whose subgroups are all normal. Please give an example. Can we give example of a finite non-abelian group with same property ?

On Symmetric Group $S_n$ and Isomorphism

I use Abstract Algebra by Dummit and Foote to study abstract algebra! At page 120, section 2 in chapter 4, there is a great result form my point of view which proves that, for any group $G$ of order $n$, $G$ is isomorphic to some subgroup of $S_n$. My question: Is there any way to […]

Frattini subgroup of a finite group

I have been looking for information about Frattini subgroup of a finite group. Almost all the books dealing with this topic discuss this subgroup for p-groups. I am actually willing to discuss the following questions : Let G be a finite group, is the Frattini subgroup of G abelian ? Why is the order of […]

Must subgroups sharing a common element be nested in each other?

Let $H$ and $K$ be subgroups of a group $G$ which have a common element besides the identity. Does this mean that either $H$ or $K$ is a subgroup of other? Let $a$ be the common element, then both subgroups contain the subgroup generated by $a$. I know it is possible for a subgroup to […]

Existence of Homomorphism so that Diagram Commutes

Let $G, H$ be groups and let $\phi: G \to H, \psi: G \to K$ be homomorphism such that $\ker \phi \subseteq \ker \psi.$ Prove that there exists a homomorphism $\theta: H\to K$ such that $\theta\circ\phi = \psi.$ Obviously one can find a homomorphism $\theta$ on $\phi(G)$ which satisfies the properties, but I don’t see […]

How to compute the automorphism group of split metacyclic groups?

I am trying to calculate the automorphism group of an affine subgroup $$G=\mathbb{Z}_p\rtimes\mathbb{Z}_{k}\leq\text{AGL}(1,p).$$ One might guess $\text{Aut}(G)=\text{AGL}(1,p)$. And this matches what I got in GAP after checking a couple examples. However, it seems proving it through by definition is messy and not particular easy. So I have been pondering a while what will be a […]