Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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0
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1answer
116 views

Group equals union of three subgroups

Suppose $G$ is finite and $G=H\cup K\cup L$ for proper subgroups $H,K,L$. Show that $|G:H|=|G:K|=|G:L|=2$. What I did: so if some of $H,K,L$ is contained in another, then we have $G$ being a union of ...
3
votes
2answers
147 views

Automorphism group of $\mathbb{Z}_p\times \mathbb{Z}_p$

How to determine the automorphism group of $\mathbb{Z}_p\times \mathbb{Z}_p$ where $p$ is a prime? Or more specifically , how to determine the element of order $2$ in this group? I got stuck here, ...
0
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0answers
54 views

How to enumerate group elements using the group generators?

I have that the group generators of the crystallographic point group 432 can be expressed in quaternion form as: $$q_A = \left[1,0,0,0\right]$$ $$q_B = \frac{1}{2}\left[1,1,1,1\right]$$ $$q_C = ...
6
votes
3answers
145 views

Example of a group which is abelian and has finite (except the $e$) and infinite order elements.

Exercise 7: Show that the elements of finite order in an abelian group $G$ form a subgroup of $G$ I just solved this exercise but I can't find example of a group which is abelian and has ...
0
votes
1answer
34 views

Quotient of abelian groups of rank $2$

Let $A, B$ be abelian groups, $B$ is contained in $A$, both $A$ and $B$ are assumed to have rank $2$. Is there a standard way to show that the quotient group $A/B$ is finite? I think there exists some ...
7
votes
2answers
204 views

If $f\colon G\to H$ is a surjective homomorphism, then $|C_G(g)| \geq |C_H(f(g))|$

Let $G$ be finite, $f\colon G\to H$ be a surjective homomorphism (hence $H$ is finite) and $g \in G$. Prove the order of center of $g$ in $G$ is greater than or equal to the order of the center of ...
1
vote
1answer
118 views

Determining the group homomorphism in semidirect product

We know that if $N$ is a normal subgroup, $H$ is a subgroup, and $\varphi$ is the group homomorphism such that $\varphi:H\to$Aut$(N)$. And this gives a unique group, called the outer semidirect ...
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3answers
292 views

Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.

Let $G$ be a finite and non-abelian group. How do I prove the following statement? $$\exists a,g,h\in G \colon\quad h=aga^{-1},\ g\neq h ,\ gh=hg.$$ Thanks in advance.
2
votes
1answer
48 views

Number of Sylow bases of a certain group of order 60

We have $G=\langle a,b \rangle$ where $a=(1 2 3)(4 5 6 7 8)$, $b=(2 3)(5 6 8 7)$. $G$ is soluble of order 60 so has a Hall $\pi$ subgroup for all possible $\pi\subset\{2,3,5\}$. $\langle a\rangle$ is ...
6
votes
1answer
141 views

Degree of transitive constituents is odd implies $|G|$ is odd

I want to prove that: the order of a permutation group $G \le S^\Omega$ is odd if and only if the degrees of all transitive constituents of $G$ and the degrees of all transitive constituents of each ...
1
vote
1answer
267 views

A counterexample of normal subgroup with cyclic Sylow 2-subgroup

We know that when a group $G$ has order $2^k m$, where $m$ is an odd integer, $G$ should have a normal subgroup with order $m$ from here. When $k=1$, this implies the index of the normal subgroup is ...
0
votes
1answer
59 views

Why don't we consider non-units as quadratic residues?

Is there any specific reason in not including non-units of $\mathbb{Z}_n$ as quadratic residues? As an examples, we say that in $\mathbb{Z}_8$, the set of quadratic residues is just {1} and not {1,4}. ...
1
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2answers
69 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 ...
2
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0answers
54 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
81 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 ...
5
votes
0answers
394 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 ...
3
votes
2answers
233 views

Finite group is generated by a set of representatives of conjugacy classes.

Could you tell me how to prove that a finite group is generated by a set of representatives of conjugacy classes? I've read this ...
1
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2answers
761 views

Show groups of symmetries of a cube and a tetrahedron are not conjugate in isometry group.

I've shown that the symmetry group of a cube and a tetrahedron are both isomorphic to S4, but I am now trying to show that they are not conjugate when considered as subgroups of isometries of 3D ...
4
votes
1answer
112 views

Group homomorphisms into a field

Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
0
votes
2answers
119 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 ...
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3answers
554 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
84 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
121 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
44 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 ...
4
votes
1answer
109 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 ...
3
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1answer
47 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
1
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1answer
352 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.
5
votes
3answers
153 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 ...
7
votes
3answers
157 views

Homomorphism from $\mathbb{Z}/n\mathbb{Z}$

Does there exist a non-zero homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $\mathbb{Z}$ ? If yes, state the mapping. How is this map exactly?
6
votes
2answers
81 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 ...
3
votes
2answers
135 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
97 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
186 views

Classify the abelian groups of order 81, 144 and 216

Classify the abelian groups of order n, 2n, 4n
2
votes
2answers
149 views

$H$ must contain every Sylow $p$-subgroup of $G$

Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that $H$ must contain every Sylow $p$-subgroup of $G$.
4
votes
2answers
273 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
929 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 ...
2
votes
2answers
111 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
160 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 ...
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0answers
92 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 ...
8
votes
1answer
123 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
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1answer
73 views

An question of $\pi$-groups for the alternating group $A_5$

Let $G=A_5$, the alternating group of degree 5. Please help me show that for $\pi = \{2,3\}$ we have that $M \leq G$ is a maximal $\pi$-group if and only if $M \cong A_4$ or $M \cong S_3$ (the ...
1
vote
2answers
273 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, ...
8
votes
1answer
231 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
51 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
61 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 $G$.
2
votes
1answer
97 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 ...
10
votes
1answer
227 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
83 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 ...
3
votes
1answer
52 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 ...
15
votes
2answers
330 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 ...