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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638 views

Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian [duplicate]

Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
7
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1answer
257 views

How do combinations (not permutations) relate to group theory?

First question. I'm just generally curious about combinations in group theory. How do they relate? If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
3
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0answers
74 views

Induction from trivial representation and number of irreducibles

Let $G$ be a finite group, $S$ a subgroup and $K$ a field, whose characteristic does not divide the group order. Let $\pi$ be the trivial representation of $S$. Are there criteria in how many ...
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2answers
89 views

Show that every subgroup of $Q_8$ is normal.

Show that every subgroup of $Q_8$ is normal. Is there any sophisticated way to do this ? I mean without needing to calculate everything out.
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0answers
432 views

Why is the Monster group the largest sporadic finite simple group?

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
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5answers
73 views

Show that If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$

If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$. I am sort of stuck with this proof and I would appreciate a hint (not a full solution, please!). Preferably, ...
2
votes
2answers
146 views

How many subgroups of $\Bbb Z_5\times \Bbb Z_6$ are isomorphic to $\Bbb Z_5\times \Bbb Z_6$

I am trying to find the answer to the question in the title. The textbook's answer is only $\Bbb Z_5\times \Bbb Z_6$ itself. But i think like the following: Since 5 and 6 are relatively prime, $\Bbb ...
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2answers
1k views

Let $N$ be a normal subgroup of index $m$ in $G$. Prove that $a^m \in N$ for all $a \in G$.

I'm trying to understand this proof: Let $N$ be a normal subgroup of index $m$ in $G$. Prove that $a^m \in > N$ for all $a \in G$. Proof $\;\;$ Let $a\in G$. Since $[G:N]=m$, then $|G/N|=m$. ...
6
votes
3answers
191 views

Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$

I'm trying to prove that Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$ (p prime). I know that Aut $\mathbb Z_p$ has $p-1$ elements because $\mathbb Z_p$ has $p-1$ possiblities of generators, so intuitively ...
3
votes
5answers
311 views

What is the largest order among all cyclic subgroups of $\Bbb Z_6\times \Bbb Z_8$?

I have a question that saying "What is the largest order among the order of all cyclic subgroups of $\mathbb{Z}_6\times \mathbb{Z}_8$?". The answer of it says that it is ...
2
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1answer
301 views

How to find all subgroups of a direct product?

I am wondering how do we find all subgroups of a direct product? Is there a method to find it? For example, how can we find all the subgroups of $\mathbb{Z}_2\times\mathbb{Z}_2$? There is the ...
4
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1answer
130 views

A question about finite simple groups

Let ‎$G‎‎$ ‎be a‎ ‎finite ‎simple ‎group. ‎Is ‎it ‎possible ‎to ‎find ‎two ‎‎ distinct ‎proper non-trivial ‎subgroups ‎‎‎$H_1‎‎$ and ‎‎‎‎$‎‎H_2$ ‎of ‎‎$G‎‎$ ‎‎ such ‎that ‎‎$\langle H_1 , ...
9
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1answer
225 views

Covering a group with the conjugates of two subgroups related by an automorphism

Let $G$ be a finite group and $H$ a proper subgroup. Then $G$ is not the union of the conjugates of $H$. This is a standard homework problem; Arturo gives a nice solution here. It is also not ...
0
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1answer
78 views

If G is a finite group, and $\exists !$ $H\leq G$ such that $|H| =n$, then $H \triangleleft G$.

So far this is what I'm thinking: Suppose the $G$ has exactly one subgroup of size $n$. Let $N$ be a subgroup of $G$ of size $n$. By Lagrange's Theorem we know $|N| \mid |G|$ But pretty much ...
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3answers
306 views

Why doesn't the Chinese remainder theorem contradict the Fundamental Theorem of Finitely Generated Abelian Groups?

I am finding a contradiction between those two theorems and I do not know what I am doing wrong. First theorem is: The group $\Bbb Z_{m_1} \times \Bbb Z_{m_2} \times \dotsm \times \Bbb Z_{m_n}$ is ...
0
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1answer
76 views

Projective linear group - solvable

Let $q\geq 5$ and let PGL(2,q) be the projective general linear group. Question Do there exists a $q$ such that PGL(2,q) is solvable?
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2answers
191 views

Other proofs that subgroups of $A_5$ have order at most 12

How can it be proved that any subgroup of $A_5$ has order at most 12? This is [Herstein, Problem 2.10.15], which also gives the hint that I can assume the result of the previous problem that $A_5$ ...
3
votes
2answers
332 views

Classifying groups of order 90.

Since $3\cdot 3\cdot 2\cdot 5=90$, we know that we have a $3$-Sylow subgroup $P_3$ of order $9$, a $2$-sylow subgroup $P_2$ of order $2$, a $ 5$-Sylow subgroup $P_5$ of order $5$. I know that $P_5 ...
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4answers
82 views

How do I show that $N\leqslant Z(G)$ without using Sylow theorems?

Let $G$ be a group of order $p^2$ ($p$ being a prime) and $N$ be a normal subgroup of $G$ such that $O(N)=p.$ Without using the fact any group of order $p^2$ is abelian or sylow theorem how to show ...
6
votes
2answers
327 views

What are central automorphisms used for?

A central automorphism is an automorphism $\theta$ for which $x^{-1}\theta(x)\in Z(G)$ for each $x\in G$. It's not difficult to prove that the set of central automorphisms forms a subgroup of ...
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5answers
447 views

Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)

I want to show without using Sylow theorem that Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime) My attempt: Since $|G|=2p,$ even $\exists~a\neq e\in G$ such that ...
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1answer
69 views

How are these infinite classes of groups of orders $6n$ and $8n$ called?

In the book Gordon James, Martin Liebeck: Representations and Characters of Groups the following three classes of groups are given in a series of exercises, where the reader is asked to find all their ...
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2answers
221 views

$p$-Group as union of subgroups

It is well known that a group can not be union of two proper subgroups. For finite $p$-groups, we can say more: A finite $p$-group can not be union of $p$ proper subgroups. Moreover, ...
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votes
2answers
661 views

If every element has prime power order and $Z(G) \neq 1$ then $G$ is a $p$-group.

In a finite group $G$ if every element is of some prime power order (prime may vary with element) and if $G$ has non trivial center then prove that $G$ is actually of prime power order. Deduce that ...
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1answer
92 views

Euler's formula and subgroups of $\mathbb Z_n$

Prove that in $\mathbb Z/n\mathbb Z$ for every divisor $d$ of $n$ there is a unique subgroup of order $d$ using the following results: $\sum_{d\mid n}\varphi(d)=n$ and the number of generators of ...
2
votes
2answers
121 views

Number of conjugacy classes of the reflection in $D_n$.

Consider the conjugation action of $D_n$ on $D_n$. Prove that the number of conjugacy classes of the reflections are $\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if } ...
8
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1answer
137 views

on finite abelian groups

Let $G$ be a finite abelian group and let $M(G)$ be the set of all elements of $G$ that fix with any automorphism of $G$. Then prove $$M(G)=\langle1\rangle \text{ or } Z_{2}$$ Attempt: We know that ...
8
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1answer
92 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
2
votes
2answers
304 views

Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
2
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4answers
79 views

Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism.

I'm stuck on this proof. I need to prove: Let $\langle a \rangle $ be a cyclic group of order $n$ and let $m$ and $n$ be relatively prime. Show that $f(x) = x^m$ is an automorphism. And this is the ...
2
votes
1answer
58 views

About finite $p$-group finitely generated.

Let $p$ a prime number. Let $G$ be a $p$-group finitely generated, say, $G = \left<x_1, x_2, \ldots, x_m \right>$. Denote by $\gamma_i(G)$ the $ith$ term of the lower central series of $G$. ...
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2answers
69 views

Question about Sylow's theorems and a particular group of order 60

I have a finite group $G$ with the following data: Its order |$G$| is 60, it has exactly 6 Sylow-5-subgroups $P_i$$\ $ (i=1,...,6) and |$N_G(P_i)$|=10 $\forall$ i. I have the following questions: ...
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3answers
83 views

Show that A is a field which has $9$ elements

For $A=\left\{\left( {\begin{array}{*{20}{c}} a&b\\ { - b}&a \end{array}} \right) | a,b\in\mathbb{Z/3Z}\right\}$ . Show that A is a field which has $9$ elements . $(A^*, .)$ is a cyclic group ...
5
votes
1answer
114 views

Induction from normal subgroup, problem with degrees

Suppose $K$ is an arbitrary field, $G$ a finite group and $N$ a normal subgroup. If one know all the irreducible representations of $N$ and then form the induced representations, one can use Mackey ...
5
votes
1answer
198 views

Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
3
votes
1answer
196 views

Orders of centralizers $C_G(g)$ in a group of order 60?

Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$? By considering ...
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3answers
113 views

$\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$.

The question is this: Prove that $\mathrm{Aut}(\mathbb Z_{11})$ is isomorphic to $\mathbb Z_{10}$. I tried to construct a mapping from $f\colon\mathbb Z_n\to \mathbb Z_n$ and $f([k])=[ka]$ where ...
1
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1answer
329 views

Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by ...
5
votes
1answer
776 views

Why is $S_5$ generated by any combination of a transposition and a 5-cycle?

Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
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1answer
113 views

Right cosets of $K=\{1,17\}$ in $U_{32}$

How can we list the distinct right cosets of $K=\{1,17\}$ in $U_{32}$, the set of positive integers relatively prime to $32$?
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3answers
121 views

Show that for G be a abelian group, $g \in G, g^m=1$, where $m=|G|$

I am pretty sure there is a name for the following theorem, but unfortunately, I don't known it. Theorem. Let $G$ be a abelian group with $m=|G|$, than for any $g \in G, g^m=1$. Currently I don't ...
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2answers
244 views

irreducible representation of non-abelian p-group

Can someone help with the following problem? Let $G$ be a non-abelian group of prime-power order $p^n$ and $E$ be an irreducible $G$-space over $\mathbb{C}$ giving a faithful representation of $G$. ...
4
votes
2answers
60 views

Existence of element of order $l$ dividing the order of the group

In this post: Order of kernel of a homomorphism , someone say that since $l$ divides $m$, we can say that there exists some element $x \in \ker (\varphi)$ such that $o(x)=l$. But why is it true? I ...
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1answer
65 views

How I can prove tht $T$ is isomorphic to a finite set of natural numbers?

Let $T$ be a finite abelian group. We can consider $T$ the as group $ℤ/nℤ$ or $ℤ/qℤ×ℤ/mℤ$. My question is: How I can prove tht $T$ is in bijection with a finite set of natural numbers? That is, I ...
0
votes
2answers
61 views

“Lifting the centralizer”

Let $G$ be a finite group, $T\le G$ and $N\unlhd G$ with $(|N|,|T|)=1$. Clearly $T$ acts by conjugation on $G$ and $N$ is a $T$-invariant subgroup; for this reason $T$ induces naturally an action on ...
5
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2answers
221 views

Upper bound for the sum of the orders in a finite group

This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution. Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
4
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2answers
1k views

A group of order 30 has a normal 5-Sylow subgroup.

There are several things that confuse me about this proof, so I was wondering if anybody could clarify them for me. Lemma Let G be a group of order 30. Then the 5-Sylow subgroup of G is normal. ...
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2answers
694 views

Orders of elements in cyclic groups

I think I'm a bit confused about the order of elements in cyclic groups. If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, ...
0
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2answers
90 views

Finite Group soluble $\iff G^{(k)}=\{e\}$ for some $k$

I am trying to understand the proof of the following proposition: A finite group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$, where $G^{(0)}=G$, $G^{(i+1)}=[G^i G^i]$ is the derived series. ...
4
votes
2answers
102 views

Frattini Subgroup of p-Groups

Letting $P$ be a $p$-group and $\Phi(P)$ be the Frattini subgroup of $P$ (the intersection of all maximal subgroups), the challenge is "Prove that $P/N$ is elementary abelian implies $\Phi(P)≤N$" ...