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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148 views

Suppose that $G$ is a finite cyclic group. Let $m=|G|$. Assume that $m\ge3$. Let $S=\{a\in G:|a|=m\}$. Prove that the cardinality of $S$ is even.

Yeah, NO IDEA WHERE TO GO FROM HERE. $|S|$ has to be equal to $2k$ with $k$ being a positive integer, but someone please offer a hint on how I can get started on this.
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1answer
47 views

How is, in this example, G isomorphic to a subgroup T in A(S)?

I'm pursuing my first introductory course in Algebra right now, and the book I'm using is Topics in Algebra by I.N. Herstein. In a segment about group theory he talks about applications of the fact ...
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1answer
124 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< ...
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2answers
215 views

Which elements of $S_8$ are in the subgroup of rigid motions of a cube?

Let the set $S\colon= \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$. Then which permutations of $S$ will appear in the group of rigid motions of a cube, which is a subgroup of $S_8$, the symmetric group on 8 letters? ...
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2answers
139 views

Is the classification of finite groups not a bit arbitrary?

I've never been able to find any details on what exactly decides what the classifications ought to be for finite simple groups. We have: Cyclic groups Alternating groups Groups of Lie type Sporadic ...
2
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1answer
110 views

About centralizers of groups

Suppose $G$ is a finite non-abelian group. Is it true that if $32 \nmid |G|$, then $G$ has at least one abelian centralizer? Thanks in advance.
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0answers
108 views

Lower bound for the order of a group element

I would like to know how to find a lower bound of an element in a large group. Let's say I have an element $x\in G$ and that $|G|=N$ is very large, say $\mathcal{O}(10^{300})$. To find the order of ...
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1answer
116 views

Group isomorphism from subgroup of $U(n)\times \mathbb{Z}_n$ to $D_n$ the dihedral group of order 2n.

I have a group $G_n = U(n)\times \mathbb{Z}_n$ with the operation $(a,x)(b,y) = (ab,ay+x)$ and I have a subgroup $H_n = \{(a,b) \in G_n | a = \pm 1\}$ which I want to show is isomorphic to $D_n$ the ...
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2answers
112 views

The norm map in group cohomology is an isomorphism if $M$ is a projective $G$-module

This is exercise III$.1.1(c)$ in Brown's "Cohomology of Groups." Let $G$ be a finite group, $M$ a $G$-module, and $\overline{N}:M_G\to M^G$ the norm map defined by $[m]\to Nm$, where $N=\sum_{g\in ...
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1answer
503 views

Subgroups and quotient groups of finite abelian $p$-groups

Motivation The fundamental theorem of finite abelian groups gives us a concise description of the isomorphism types of finite abelian $p$-groups $G$ (in the following, $p$ is a fixed prime). The ...
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1answer
49 views

Groups with no abelian centralizer

Suppose $G$ is a finite group with no abelian centralizers. Is it true that $G$ must be a 2-group? Thanks for any help.
2
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1answer
79 views

Example of a finite solvable group with non-solvable automorphism group

If $G$ is a finite group with $\text{Aut}(G)$ solvable, then $G$ itself must be solvable. Because then $G/Z(G)$ is solvable and obviously $Z(G)$ is solvable, and hence $G$ itself must be solvable. But ...
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0answers
15 views

Cyclic Groups of Infinite and Finite order [duplicate]

If a cyclic group has an element of infinite order , how many elements of finite order does it have ?
3
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3answers
108 views

Combinatorics inside of $GL(n,q)$

I'm studying conjugacy classes of subgroups of $GL(n,q)$ of the form $(\mathbb{Z}/p\mathbb{Z})^r$ where $q=p^d$ and $r$ is some non-negative integer. I've been able to show that for $n=p=2$ and for ...
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2answers
49 views

Uniqueness of Inverses in Groups Implies Associativity Holds?

I was checking multiplication tables for groups with 4 elements, to see which tables "passed" the group axioms of closure, associativity, identity and inverses. But then I had a question, so hopefully ...
2
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1answer
174 views

Classification of transitive G-sets for a given group of small order

Given a group of small order (<30), how does one go about systematically finding all the transitive G-sets up to isomorphism? By X and Y being isomorphic we mean there are maps $f:X \rightarrow Y$ ...
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2answers
63 views

What must be true about $n$ for there to be zero divisors in $\mathbb{Z_n}$

Precisely,what must be true about $n$ for there to be zero divisors in $\mathbb{Z_n}$ (i.e. elements $[a]_n$ and $[b]_n$ such that $[a]_n[b]_n=[0]_n$ but $[a]_n,[b]_n\not=[0]_n$? State your theorem as ...
4
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1answer
122 views

On nilpotent factor group

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ with the property that $G/N$ is nilpotent. Prove that there exists a nilpotent subgroup $H$ of $G$ satisfying $G = HN$. This is ...
2
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1answer
150 views

At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic [closed]

Suppose we have a finite group $G$ of finite order $n$. For every $d\mid n$, $G$ has at most one subgroup of order $d$. Show that $G$ is cyclic.
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0answers
35 views

a exercise in Berkovich‘ book

in Berkovich' book characters of finite groups there is a exercise in page 59. exercise 15, a group G is a Q-group if and only if for any cyclic subgroup Z of G who can tell me how to prove it ? ...
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1answer
63 views

An element of order $n$ generates a normal subgroup of $D_n$

Let $a$ be an element of order $n$ of $D_n$. Show that $\langle a\rangle \lhd D_n$ and $D_n/\langle a\rangle \cong \mathbb Z_2$. Proof: Let $K = <a>$ for some a ∈ G. Let H ≤ K be an ...
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1answer
50 views

Non-isomorphic product of two groups

I know this is a simple question, but I'm not able to reason it out right now. Why is $\pm I_n \not\cong \pm I_n \times \pm I_n$?
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2answers
129 views

Elements of order $10$ in $\Bbb Z_2 \times \Bbb Z_{10}$

How many elements in the group $\mathbb Z_2 \times \mathbb Z_{10}$ are of order $10$? I think the easiest way to answer this question might be to write them out, but I'm not sure how to write ...
3
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1answer
223 views

List all cosets of H and K

Let $G = \mathbb Z_3 \times \mathbb Z_6$, $H = \langle (1,2)\rangle$ and let $K = \langle (1,3)\rangle$. List all cosets of $H$ and $K$. Can somebody please explain me how to do this problem. ...
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2answers
66 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 ...
0
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1answer
62 views

Find the number of cosets$ [G:H] $?

Assume that $G$ is a cyclic group of order $n$, that $G =\ <a> $, that $k|n$ , and that $H=<a^k>$. Find $[G:H] $ the number of cosets to the subgroup H I think that since $k|n$ ...
3
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2answers
161 views

How many elements are there in the intersection of two subgroups of a finite cyclic group?

Let's assume that we have two subgroups $H_1$ and $H_2$ in $\mathbb{Z}_n$ with $k$ and $l$ elements respectively. How many elements are there in the intersection $H_1\cap H_2$? Let denote this by ...
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2answers
511 views

Is it possible to have a non-trivial homomorphism of some finite group into some infinite group?

Let $G$ be a group of some finite order, and let $G^\prime$ be some group of infinite order. Then there is the trivial homomorphism of $G$ into $G^\prime$ which maps each element of $G$ into the ...
2
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1answer
171 views

When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?

Let $m$ and $n$ be two given positive integers such that $m<n$. Then what are the necessary and sufficient conditions for the groups $(\mathbb Z_m,+_m)$ and $(\mathbb Z_n,+_n)$ to be homomorphic ...
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3answers
464 views

If a group $G$ has odd order, then the square function is injective.

Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this ...
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2answers
288 views

Show there exists a fixed positive integer $n$ such that $a^n = e$ for all $a\in G$.

Let $G$ be a finite group. Show there exists a fixed positive integer $n$ such that $a^n = e$ for all $a\in G$. We know: $n$ is independent of $a$.
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1answer
118 views

Finding the permutation that shows two permutations are conjugates method?

Problem: Given $\sigma=(12)(34)$ and $\gamma=(56)(13)$ find $\tau\in S_6$ with $\tau^{-1}\sigma\tau=\gamma$ Attempt: I'm kind of new to this but from what I understanding find $\tau$ that satisfies ...
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1answer
177 views

Could we reasonably classify finite groups?

So I have been reading some work on $p$-groups, and I noticed a particularly disturbing sentence: "There is no hope of finding a finite set of invariants that will define every p-group up to ...
18
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1answer
460 views

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
2
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1answer
72 views

Generators of $S_n$

Show that $S_n$ is generated by the set $ \{ (12),(123\dots n) \} $. I think I can see why this is true. My general plan is (1) to show that by applying various combinations of these two cycles you ...
3
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2answers
267 views

Finite abelian group with common nontrivial subgroup $H_0$ is cyclic.

let $G$ be a finite abelian group such that it contains a subgroup $H_{0}\neq (e)$ which lies in every subgroup $H\neq(e)$, prove that $G$ is cyclic. what is order of $G$? i try to solve this problem, ...
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1answer
55 views

When are these two groups isomorphic?

Let $m$, $n$, $p$, $q$ be positive integers such that $mn = pq$, gcd($m,n$) $\ne 1$, gcd($p,q$) $\ne 1$, and lcm($m,n$) $=$ lcm($p,q$). Then under what condition(s) can the groups $Z_m \times Z_n$ and ...
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3answers
152 views

What is a good way to think of Factor Groups?

I'm having a hard time thinking about factor groups. I just don't understand what notation like $\mathbb{Z}_{60}/\langle 12 \rangle$ means. Furthermore, when asked about giving the order $26 + \langle ...
0
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1answer
75 views

Automorphism of $Q_8$ [duplicate]

Is there anyone could help me to prove that $Aut(Q_8)=S_4$? Someone told me that there's an isomorphism between the rigid motions of cube and $Aut(Q_8)$, any ideas? Thank you!
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1answer
769 views

Order of subgroups and number of elements of order $3$ in a group of order $9$

Let $G$ be a group of order $9$. 1) State the possible orders of subgroups and elements in $G$. 2) Find the number of elements of $G$ of order $3$ in the cases where (a) $G$ is ...
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1answer
112 views

Every nontrivial subgroup $H$ of $S_9$ containing some odd permutation contains a transposition. [duplicate]

This is a true or false question. Apparently, it is false, but I don't follow. Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it ...
2
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1answer
34 views

Does the induced representation always contain a non-trivial representation

Let $H$ be a proper subgroup of a finite group $G$ - not normal. Does $Ind_H^G 1$ contain a non-trivial representation? The Frobenius character formula was my original approach, but I can't rule out ...
2
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1answer
113 views

How to calculate an index $(G:U)$

I have the group $G = \def\Z{\mathbb Z}\Z/9\Z$ and the subgroup $U = \{\bar 0,\bar 3,\bar 6\} \subseteq G$. My first question is, what mean the lines over the elements of the subgroup. And how can ...
2
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1answer
136 views

Calculate the Factor Group: $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$

I am attempting to understand and compute: $(\mathbb{Z}_4 \times \mathbb{Z}_6)/\langle(0,2)\rangle$ I know $(0,2)$ generates $H = \{(0,0),(0,2),(0,4)\}$, which has an order of 3 because there are 3 ...
3
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1answer
373 views

Without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$.

Prove the following without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$. So this proof is pretty trivial if you have Lagrange's theorem, but ...
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1answer
502 views

A subgroup of $S_n$ contains only even permutations or half of them are even. [closed]

Show that for every subgroup $H$ of $S_n$ for $n \geq 2$, either all the permutations in $H$ are even or exactly half of them are even.
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56 views

correspondence theorem question

If $G$ has order $12$ and $G'$ has order $6$, produced by elements $x$ and $y$ respectively, and $\phi$ maps $G$ to $G'$ which is defined by $\phi(x^n)=y^n$, how is the correspondence exhibited in the ...
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0answers
35 views

What are the cosets of this presentation?

I'm reading a book on algebra, and they give a presentation for $S_3$, with 6 elements $\{1, x, x^2, y, x y, x^2y\}$ as $$x^3 = 1,\quad y^2 = 1,\quad y x=x^2y$$ Now later in the book, there is a ...
2
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1answer
234 views

Subgroups of $U(n)$ isomorphic to a direct sum of cyclic groups

Let $U(n)$ to be the set of all positive integers less then $n$ and relatively prime to $n$. Then $U(n)$ is a group under multiplication modulo $n$. A. Find integer $n$ such that $U(n)$ contains ...
7
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1answer
142 views

character degree and solvability

There is an unsolved problem in Berkovich's book "Characters of Finite Groups Part 2" I state here: Is $G$ solvable if $\chi(1)^2$ divides $|G|$ for all $\chi \in \operatorname{Irr}{(G)}$? Can ...