2
votes
1answer
49 views
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
using Cauchy's theorem proof is easy but how can I do this without using this?
1
vote
2answers
40 views
Give an example of the $a,b,c$ which satisfies conditions in the generating set
How to derive the specific case of the generating element of a group given its generating set. For example, when
$$G=\langle a,b,c|a^3=b^3=c^2=1,ab=ba,ca=a^2c,cb=b^2c\rangle$$
we can let $G\subset ...
1
vote
0answers
17 views
Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)
This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups.
More precisely, given discrete groups below (a), (b), (c):
...
3
votes
1answer
45 views
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
3
votes
0answers
52 views
Discrete subgroups of SU(n) and SO(n).
Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.
I would like to know whether there is a complete understanding of discrete ...
0
votes
2answers
57 views
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to ...
4
votes
3answers
82 views
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.
Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at
least two elements.
We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$.
If $G$ is ...
3
votes
1answer
61 views
Infinite products of a (finite) group
So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say ...
4
votes
2answers
94 views
Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any ...
8
votes
2answers
35 views
The index of $\xi_4^*$ in $\xi_4$
Just seeing if i'm right:
With the set of solutions for $z^4=1$: $\xi_4=\{1,i,-1,-i\}$, one can construct the group of the $4$th roots of unity: $(\xi_4,\cdot_\mathbb{C})$ and its multiplicative ...
-3
votes
0answers
39 views
Show there does not exist $\alpha$ in $ S_4$ s.t. $ (1 2)(3 4)\alpha = \alpha(1 2 3 4)$ [duplicate]
Show that there does not exist a permutation $\alpha \in S_4$ such that $ (1 2)(3 4)\alpha = \alpha(1 2 3 4)$.
3
votes
1answer
46 views
sylow basis of finite solvable groups
Let $G$ be a finite solvable non-$p$-group and $A$ be a maximal subgroup of $G$.
Therefore $A$ is of primary index $p^{n} $, that is $|G : A|=p^{n}$ where ...
1
vote
1answer
60 views
Non trivial Automorphism [duplicate]
Prove that every finite group having more than two elements has a nontrivial Automorphism.
It is from Topics in Algebra by Herstein. I am not able to solve.
6
votes
3answers
78 views
Finding the subgroup of $(\mathbb Z_{56},+)$ which is isomorphic with $(\mathbb Z_{14},+)$ by GAP
I am sorting some easy questions for the students in Group Theory I. One of them is:
Is $(\mathbb Z_{14},+)$ isomorphic to a subgroup of $(\mathbb Z_{35},+)$? What about $(\mathbb Z_{56},+)$?
I ...
5
votes
3answers
99 views
Homomorphism from $\mathbb{Z}/n\mathbb{Z}$
Does there exist a homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
6
votes
2answers
66 views
Embeddings of $GL(n-1,q)$ into $GL(n,q)$
Let $n>2$ and $q$ be a prime power. I want to find all the embeddings of $GL(n-1,q)$ into $GL(n,q)$. An embedding I first thought of is as following. Let $V$ be an $n$-dimensional vector space over ...
2
votes
2answers
48 views
Rotman's Introduction to to the theory of groups. Exercise 3.45.
Can you give me a hint on the first part of the exercise?
Let $p$ be a prime and let $X$ be a finite $G$-set, where $|G| = p^n$ and $|X|$ is not divisible by $p$. Prove that there exists $x \in X$ ...
8
votes
1answer
60 views
Cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$?
What are the cyclic subgroups of $GL_2(\mathbb{Z}/p\mathbb{Z})$, the general linear group over the finite field of order $p$, where $p$ is prime?
Obviously, each cyclic subgroup is generated by some ...
0
votes
1answer
62 views
1
vote
2answers
75 views
$H$ must contain every Sylow $p$-subgroup of $G$
G is finite and has a prime factory. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that
4
votes
2answers
60 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
7
votes
1answer
73 views
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that the order of the center of $G$ is 1 or $pq$.
Let $G$ be a group of order $pq$, with $p$ and $q$ prime. Prove that
the order of the center of $G$ is 1 or $pq$.
Let me start off with what I did:
Assume $G$ is abelian. Then we know ...
3
votes
3answers
77 views
Sum of two squares in a $\Bbb Z/p\Bbb Z$
I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of ...
1
vote
1answer
38 views
$P \cap N$ is a Sylow $p$-subgroup of $N$, where $N$ is normal in $G$ and $P$ is a Sylow $p$-subgroup of $G$?
In 'A Course in Group Theory' by Humphreys, Proposition 11.14 says that if $G$ is a finite group, $P$ is a Sylow $p$-subgroup of $G$ and $N$ is a normal subgroup of $G$, then $P \cap N$ is a Sylow ...
0
votes
0answers
47 views
Generators in $p$-groups
Let $G$ be a non-abelian finite $p$-group such that $G/[G,G]$ is elementary abelian group of order at least $p^3$. Let $S=\{\bar{x_1},\bar{x_2},\bar{x}_3,\cdots,\bar{x}_n\}$ be independent generators ...
7
votes
1answer
66 views
Why does the automorphism used to construct the group have to be non-inner?
I have a question on why a particular assumption is made that the automorphism used to construct a certain group be non-inner.
In [Herstein, Topics in Algebra, p. 69], a construction of a nonabelian ...
1
vote
2answers
90 views
Multiplicative group of integers modulo n
Consider the abelian group $U_n=\{a\in \mathbb{Z}_n:(a,n)=1\}$. Is there a natural way to understand it as a subgroup of any other interpretation of the cyclic group of order $n$. For example, ...
3
votes
0answers
71 views
How to recover the integral group ring?
I would like to solve the following exercise:
Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
3
votes
0answers
42 views
Any finite subgroup of the abelian group $(F-\{0\},\cdot)$ is cyclic? ($F$ a field) [duplicate]
I found this problem:
Suppose that $F$ is a field, and that $(F-\{0\},\cdot)$ is an abelian group. Show that if $H$ is a finite subgroup of $F-\{0\}$, then $H$ is cyclic.
What I have done is:
...
2
votes
1answer
52 views
About commutators and center o a certain group
Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.
3
votes
1answer
49 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 ...
9
votes
1answer
189 views
What can we say about the size of $HK\cap KH$ when $HK\neq KH$?
If $G$ is a finite group, and $H$, $K$ are proper subgroups of $G$, then it is not necessary that $HK=KH$. But, these two subsets have same size. The question I would like to ask, then, is
If ...
2
votes
4answers
70 views
What is the meaning of the following quotient group?
$S_4 / \{(1 \, 2)(3 \, 4), (1), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \}$
We let $H=\{(1 \, 2)(3 \, 4), (1), (1 \, 3)(2 \, 4), (1 \, 4)(2 \, 3) \}$ be a subgroup. Then the above notation is the set of ...
2
votes
1answer
25 views
Sharply t-transitive groups.
Suppose that $G$ acts sharply $t$-transitively on a set $\{1,\cdots, n\}$. Then I want to show that if $n = t + 2$ then $G = A_n$.
I can indeed show this, but I feel it's unnecessarily messy. My way ...
9
votes
2answers
101 views
Which finite groups are the group of units of some ring?
Motivated by this question:
Which finite groups are the group of units of some ring?
Which finite groups are the group of units of some finite ring?
Which finite abelian groups are the group of ...
4
votes
1answer
58 views
$D_8$ as a derived subgroup
Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them ...
0
votes
1answer
50 views
Abstract orbits stabilizers
Consider $D_8$ acting on itself by conjugation. Find orbits and stabilizers for all elements of $D_8$.
$$D_8=\{1,r,r^2,r^3,b,br,br^2,br^3\}$$
So far I have the orbits: {$1$}, {$r^2$}, {$r,r^3$}, ...
0
votes
2answers
54 views
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
3
votes
1answer
37 views
Dihedral groups and normal subgroups
Consider
$D_8$ ={$1,r, r^2, r^3,b,br,br^2,br^3$} and the subgroup, $H$={$1,r^2$} and $K$={$b$} of $D_8$
I really need some help with these particular problems.
Show that $H\lhd$$D_8$, but ...
1
vote
2answers
87 views
Can you find an isomorphic group?
Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
2
votes
2answers
101 views
Extending a homomorphism $f:\left<a \right>\to\Bbb T$ to $g:G\to \Bbb T$.
Suppose $G$ is an abelian group and $a\in G$ and
$$f:\left<a \right>\to\Bbb T$$
is a homomorphism. Can $f$ be extended to a homomorphism on $G$:
$$g:G\to \Bbb T$$
?
$\Bbb T$ is the circle ...
0
votes
4answers
113 views
Can a group of order 3000 be a simple group?
Can a group of order 3000 be a simple group? How about the case of a group of order 1000?
1
vote
2answers
43 views
Let H and K be subgroups of the finite group G and supposes $|H|^{2}>|G|$ and $|K|^{2}>|G|$. Prove $H \cap K$ has at least two elements
So I supposed $|H \cap K|>1$
$\Rightarrow |HK||H \cap K|> |HK|$
Which eventually implied that
$\Rightarrow |H \cap K|>|G|$
Thus since G is a group, and H and K are subgroups then the ...
2
votes
1answer
46 views
Let G be a finite group and let H and K be subgroups of G. Suppose [G:K] and [G:H] are relatively prime. Prove G=HK [duplicate]
So I am rather confused on where to start this proof so all I've got is
$[G:H \cap K]=[G:H][H:H \cap K]$
$[G:H \cap k]=[G:K][K:H \cap K]$
Thus that implies $[G:H][H:H \cap K]=[G:K][K:H \cap K]$
...
3
votes
1answer
40 views
order of $\langle (123) , (234) \rangle$
As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ .
I've started doing all the multiplications between the elements, and I've ...
1
vote
1answer
49 views
The relationship between inner automorphisms, commutativity, normality, and conjugacy.
An inner automorphism of a group $G$ is defined to be a function $f: G \to G$ such that for $x\in G$ $f(x) = a^{-1}xa.$
I have three somewhat broad questions about this:
Why is it related to ...
3
votes
2answers
74 views
How to enumerate subgroups of each order of $S_4$ by hand
I would like to count subgroups of each order
(2, 3, 4, 6, 8, 12) of $S_4$,
and, hopefully, convince others that I counted them correctly. In
order to do this by hand in the term exam, I need a ...
0
votes
2answers
38 views
When does $S_n$ have a subgroup with order $p^2$ where $p$ is prime?
I'm attempting this homework problem, and I'm not sure where to start. Here is the problem and how what I've got so far.
Let $p$ be a prime number. What is the least positive integer $n$ such that ...
2
votes
2answers
70 views
$G$ $p$-group. If $H\triangleleft G$, then $H\cap C(G)\ne \{e\}$
I'm pretty new on this subject and I need a hint to begin to solve this question:
If $G$ is a finite p-group, $H\triangleleft G$ and $H\ne \{e\}$, then
$H\cap C(G)\ne \{e\}$
Thanks for any ...
7
votes
3answers
177 views
If $G$ has only 2 proper, non-trivial subgroups then $G$ is cyclic
Is the following true?
If $G$ has two proper, non-trivial subgroups then $G$ is cyclic.
