1
vote
3answers
65 views

I have a question about groups of finite order.

I want to show that if a group G has finite even order then it must have an odd number of elements that are their own inverse, and if G has odd order then it has no elements of order two. I know this ...
2
votes
1answer
25 views

Solving a divisibility problem using group theory

Inspired by this I decided to show this: Let $P_n=\displaystyle\prod_{1\leq i<j\leq n} \big(x_i-x_j\big)$ where $x_1\,\dots\, x_n$ are arbitrary integers. Prove $n!\,\big|\, 2P_n$. I am ...
2
votes
2answers
110 views

The center of a group with order $p^2$ is not trivial

Let $p$ be a prime and $G$ be a group of order $p^2$. Show that $Z(G)\neq 1$. Is there a proof of this nice fact that doesn't use the class equation?
3
votes
2answers
69 views

Simpler proof of $g\,h\,g^{-1} = h^a \Rightarrow g^n\,h\,g^{-n} = h^{a^n}$

In a rather easy online lecture on group theory (which included many obvious statements such as "the only divisors of a prime number $p$ are $1$ and $p$"), the professor began a proof by assuming that ...
0
votes
1answer
60 views

A nice group isomorphism

Show that $$k(\mathbb{Z}/n\mathbb{Z})\cong (\gcd(n,k)\mathbb{Z})/n\mathbb{Z}.$$ I want to see as many as possible proofs of this nice fact.
4
votes
2answers
107 views

On the existentence of an element of a group whose order is the LCM of orders of given two elements which are commutaive

I came up with the following proposition. Proposition Let $G$ be a group. Let $x, y$ be elements of finite order in $G$ such that $xy = yx$. Let $n$ be the order of $x$. Let $m$ be the order of $y$. ...
4
votes
3answers
154 views

Alternative Creative Proofs that $A_4$ has no subgroups of order 6

Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$. While I know how to prove this statement, I am ...
5
votes
0answers
127 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
2
votes
1answer
91 views

How many elements of order $7$ are there in a group of order $28$ without Sylow's theorem

How many elements of order $7$ are there in a group of order $28$ I need to prove this result without using the Sylow's Theorem.By Sylow's Theorem it has only one subgroup and the anser becomes ...
2
votes
1answer
159 views

Group theory - proof check about index and quotient group

I'm studying Cayley's Theorem on the Humpreys "A Course in Group Theory" and i did not understand a passage in a preposition. (pag 86 Corollary 9.23). It claims: "Let $H \leq G$ with finite index $n$. ...
16
votes
2answers
348 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
1
vote
4answers
78 views

How do I show that $N\leqslant Z(G)$ without using Sylow theorems?

Let $G$ be a group of order $p^2$ ($p$ being a prime) and $N$ be a normal subgroup of $G$ such that $O(N)=p.$ Without using the fact any group of order $p^2$ is abelian or sylow theorem how to show ...
2
votes
5answers
237 views

Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)

I want to show without using Sylow theorem that Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime) My attempt: Since $|G|=2p,$ even $\exists~a\neq e\in G$ such that ...
28
votes
3answers
538 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
3
votes
1answer
324 views

Prove a group of even order has an odd number of elements of order 2 using properties of cosets

I am aware that it is straightforward to prove that a group of even order has an odd number of elements of order 2 just by noting that elements of order greater than 2 come in pairs necessarily (i.e. ...
2
votes
1answer
123 views

A divisible abelian group is not finitely generated: group theoretical proof

Do you a group theoretical proof of the following result?: Theorem: A (non trivial) divisible abelian group is not finitely generated. The only proof I know uses the fundamental theorem of ...
1
vote
1answer
194 views

Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.

I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. ...