3
votes
1answer
47 views

What is $\mathbb{Z}_2 \times \mathbb{Z}_4$ isomorphic to - Fraleigh p. 112 Exercises 11.32e

(e). p. 4 of PDF - $\mathbb{Z}_2 \oplus \mathbb{Z}_4 \not\simeq \mathbb{Z}_8$. Another solution (1.) Why is $\mathbb{Z}_2 \oplus \mathbb{Z}_4$ not cyclic? Is it because of $ \gcd(2, 4) = 2 \neq 1 ...
1
vote
1answer
45 views

A group of odd order has no non-identity elements which are conjugate to their inverse.

I want some verification for my proof to a homework problem. (Is it correct? Is there a simpler way to do this?) Let $G$ be a finite group of odd order and suppose there is an element $g$ that is ...
3
votes
1answer
43 views

$S_n$ acting on $\{1\;…\;n\}\times \{1\;…\;n\}$

Let $X=\{1,\;...\;n\}$ and $S_n$ act transitively on $X\times X$ i.e. $s:\;(m,n)\mapsto (s(m),s(n))$. Compute the orbits under this action. Attempt: I claim that there are only two orbits, ...
3
votes
2answers
123 views

Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
3
votes
2answers
69 views

$x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$ [duplicate]

Problem Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order $2$ in $G$ that generate $G$. Prove that $G \cong D_{2n}$, where $\vert xy \vert = n$. Solution We have $x^2 = ...
0
votes
1answer
31 views

Prove something with finite groups and homomorphism…

$H,G$ are finite groups, and $\gcd(|H|,|G|)=1$. I need to prove that if $\varphi:G\to H$ is homomorphism it must be the Trivial Homomorphism. My try: I assume that $\varphi:G\to H$ is homomorphism. ...
1
vote
0answers
45 views

Prove that $\varphi$ is automorphism

$G$ is commutative group. $|G|=n$. $m\in \mathbb{N}$ and $\gcd(m,n)=1$. I need to prove that $\varphi :G\to G$, $\varphi(x)=x^m$ is automorphism of G. My try: I assume that $a\in \ker(G)$, so $a\in ...
0
votes
2answers
74 views

Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$

Claim: Let $G$ be a group of order $27.$ If $G$ is not abelian, then $|Z(G)|=3,$ where $Z(G) = \{z \in G: zg=gz, \forall g \in G\}$ May I know if my proof is correct? Thank you. Lemma: $G/Z(G) $ is ...
0
votes
2answers
36 views

Some problems in group theory

May I know if my proof/solution is correct? Thank you v. much. 1.) If $G, H$ are finite groups of order $10$ and $21$ respectively, then every homomorphism $f:G \to H$ satisfies $f[G] = \{e_H\}.$ ...
2
votes
1answer
158 views

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$

If $H,K$ are subgroups of finite group $G,$ then $|G:(H \cap K)|\leq |G:H||G:K|$ May I know if my proof is correct? Thank you v. much. Proof: $$|G:(H \cap K)| \leq |G:H||G:K| ...
2
votes
0answers
46 views

Results on Hall $\pi$-subgroups - proof verification

I would appreciate verification of the following proofs. The problems are all closely related and the proofs are similar, so I think it's only appropriate to post them together (also, they are part ...
1
vote
1answer
114 views

Let $p,q$ be distinct primes. Find number of generators of $(\mathbb{Z}/pq\mathbb Z, +)$

May I verify if my proof to the a/m claim is correct? Thank you. #generators $=\phi(pq)$. Let $ A = \{x\in \mathbb{N}: q\mid x \wedge x< pq\}$ and $B = \{y\in \mathbb{N}: p\mid y \wedge y< ...
0
votes
2answers
63 views

I don't think I'm using an assumption in this proof. Anything wrong?

Define the exponent $\exp(G)$ of a finite group $G$ to be the smallest positive integer $k$ such that $g^k = e$ for all $g \in G$. The question asks If $G$ is a finite abelian group, prove that ...
1
vote
1answer
130 views

Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Find all numbers $n$ such that $S_7$ contains an element of order $n.$ Identity is the only element of order $1.$So $n=1$ is possible. Case 1: Elements that can be written as a unique cycle of ...
1
vote
2answers
140 views

Proving $G\cong \Bbb{Z}_p$ if $|G|$ is prime.

For a group $G$, if $|G|$ is prime, then I have to prove $G\cong \Bbb{Z}_p$. Take any element $g\in G$. As $G$ cannot have proper subgroups, and as it also has finite order, $|g|=p$. Hence, ...
7
votes
0answers
159 views

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.

Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal. Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$. $n_2 \equiv 1 ...