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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1answer
168 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
322 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
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1answer
57 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 ...
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1answer
152 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 ...
2
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1answer
358 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 ...
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1answer
64 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}. ...
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2answers
71 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 ...
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0answers
67 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): ...
4
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1answer
97 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 ...
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0answers
570 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 ...
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2answers
357 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 ...
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2answers
1k 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
127 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: ...
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2answers
145 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
720 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 ...
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1answer
127 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 ...
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2answers
128 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 ...
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2answers
45 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 ...
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1answer
143 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 ...
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1answer
55 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 ...
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1answer
564 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.
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3answers
202 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 ...
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3answers
171 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?
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2answers
82 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 ...
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2answers
163 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
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1answer
109 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 ...
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1answer
307 views

Classify the abelian groups of order 81, 144 and 216

Classify the abelian groups of order n, 2n, 4n
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2answers
176 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 $G$.
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2answers
424 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
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1answer
1k 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 ...
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2answers
143 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 ...
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1answer
196 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
103 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
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1answer
170 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 ...
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1answer
88 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 ...
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2answers
325 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
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1answer
285 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 ...
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0answers
53 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: ...
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1answer
65 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 $G$.
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1answer
122 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 ...
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1answer
239 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 ...
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4answers
93 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 ...
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1answer
58 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 ...
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2answers
558 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 ...
7
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1answer
130 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 ...
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0answers
166 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
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1answer
70 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
230 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
137 views

Finite groups such that $H^1(G,M)=0$ for any simple $G$-module $M$

I'm trying to understand for which finite groups $G$ the augmentation ideal of $\mathbb{F}_2G$ is generated by a single element over $\mathbb{F}_2G$. I'm reading a paper with a result that implies ...
3
votes
1answer
127 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 ...