The study of symmetry: groups, subgroups, homomorphisms, actions
0
votes
1answer
21 views
Group of infinite sequences of elements in a given finite group
Could you tell me why each element of a group of infinite sequences of elements in a given finite group has finite order?
1
vote
1answer
42 views
$\mathbb Z_p^*$ is a group.
I'm trying to prove that $\mathbb Z_p^*$ ($p$ prime) is a group using the Fermat's little theorem to show that every element is invertible.
Thus using the Fermat's little theorem, for each $a\in ...
2
votes
1answer
38 views
$G$ is a nonabelian finite group, then $|Z(G)|\leq \frac{1}{4}|G|$
If $G$ is a nonabelian finite group, then I have to show that $|Z(G)|\leq\frac{1}{4}|G|$, where $|Z(G)|$ denotes the center of group.
I have got this question through random search on net. I am ...
0
votes
2answers
59 views
Either all elements of a subgroup of $\mathbb{Z}_n$ are even or exactly half of them
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb{Z}_n$.
Prove that either every member of $H$ is even or exactly half of the members of $H$ are even.
Well, I know that if $0 ...
0
votes
1answer
52 views
Which elements generate $\mathbb Z \oplus \mathbb Z$
It's a simple question, but I have a little problem with an aspect of the proof.
I'm trying to prove that this set $\{(1,0),(0,1)\}$ generates $\mathbb Z \oplus \mathbb Z$. Following the definition ...
1
vote
1answer
42 views
There exist Sylow subgroups $P$ and $Q$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.
From D&F's sylow theory section:
Show that if $n_p\not\equiv 1 \mod p^2$ then there are distinct Sylow $p$-subgroups $P$ and $Q$ of $G$ for which $[P:P \cap Q]=[Q : P \cap Q] = p$.
Are ...
2
votes
1answer
47 views
How to determine centralizers of $A_5$?
Prove that the order of a conjugacy class of $A_5$ is $1,12,15$ or
$20$.
There are 4 different cycle type in $A_5$. There are 20 permutations of cycle type $(123)$, 24 permutations of cycle ...
2
votes
2answers
73 views
Can a group of order $55$ have exactly $20$ elements of order $11$?
Can a group of order $55$ have exactly $20$ elements of order $11$? Give a reason for your answer
by sylow theorem the answer is easy but without using sylow how can I solve this.can anyone help me ...
2
votes
2answers
46 views
Prove the third isomorphism theorem
I'm trying to prove the third Isomorphism theorem as stated below
Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$
I look up for some ...
2
votes
1answer
41 views
Let $G$ a finite group. And let $\varphi : G→S(G):g↦λ_g$. If $|G|=nm$ and $|x|=n$, then $\varphi(g)$ is the product of $m$ disjoint $n$ cycles.
Let $G$ a finite group. And let $\varphi : G→S(G):g↦λ_g$. If $|G|=nm$
and $|g|=n$, then $\varphi(g)$ is the product of $m$ disjoint $n$
cycles.
When writing this question, I think I've proven ...
1
vote
1answer
27 views
List of all elements of the Weyl group of type $C_3$.
What is the list of all elements of the Weyl group of type $C_3$ in terms of simple refletions $s_1, s_2, s_3$? There are 48 elements in the group. Thank you very much.
4
votes
2answers
63 views
on a group with perfect automorphism group
A group $G$ is called perfect if $G=G'$.
Does there exist a group $G$ such that $Aut(G)$, the automorphism group of $G$, is perfect?
1
vote
1answer
27 views
Studying the action of $GL(V)$ on the vector space $V$
The statement I am trying to prove is the following.
Let $k$ a field and $V$ a $k$-vector space of finite dimension. Let
$\mathscr{B}$ be the set of ordered $k$-bases of $V$. The natural
...
0
votes
3answers
60 views
Why is $| \rm{Aut}(\mathbb{Z}_n) | = \phi(n)$?
If $G$ is a finite cyclic group of order $n$, prove that $| \rm{Aut}(G) | = \phi(n)$, where $\phi(n)$ is the Euler's totient function.
Can someone please help me with this?
1
vote
0answers
48 views
(Hypothetical) simple group of order 1004913
In 6.2.29 of Dummit and Foote, the exercise is to prove a group of order $3^3*7*13*409$ is not simple by using the permutation representation of degree 819. After having calculated the uniquely ...