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
60 views

Maximal subgroups of almost simple classical groups which are almost simple

Let $G$ be a finite almost simple group with socle $S$ classical and $M$ be a maximal subgroup of $G$ not containing $S$. I'm interested in the pairs $(G,M)$ such that $M$ is almost simple. If $M$ ...
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1answer
79 views

Show every automorphism of a specific group arises the same way.

I have a group $G_n = U(n) \times \mathbb{Z}_n$ with the operation $(a,x)(b,y) = (ab, ay+x) $ where $U(n)$ is the multiplicative group of integers modulo n and $\mathbb{Z}_n$ is the additive group of ...
5
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0answers
254 views

Show any group of order $275$ has an element of order $5$.

This is what I have. Note: I'm not allowed Cauchy's theorem or Sylow theorems. Let $|G| = 275$. So I know $275 = 5\times5\times11$. If I assume that $G$ is cyclic then there exists $x\in G$ such that ...
0
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1answer
92 views

Without using any Sylow theorem, if every element is a $p$-element then $G$ is a $p$-group

How can we prove the following theorem without using any Sylow theorem? Let $p$ be a prime. In a finite group $G$, if every element is a $p$-element then $G$ is a $p$-group. Or is it possible to ...
5
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3answers
186 views

show $\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $.

I am trying to determine if $\Bbb Z_6 \cong \Bbb Z_3 \times \Bbb Z_2 $. I noticed that $\Bbb Z_6$ has a generator $1$ and $\Bbb Z_3 \times \Bbb Z_2$ has generator $(1,1)$. Now I set up the bijection ...
0
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1answer
72 views

$G$ is a subgroup in the symmetric group $S_6$, $|G| = 18$, and has a Sylow 3-subgroup $H$. Show that $H$ is not cyclic. [closed]

I'm doing an exercise I have come across this sub-question that are causing me a lot of trouble: $G$ is a subgroup in the symmetric group $S_6$, $|G| = 18$, and has a Sylow 3-subgroup $H$. Show ...
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1answer
40 views

If $G=H\rtimes K$ where $H$ is cyclic, and $K$ is abelian, why is $G$ abelian?

This is a curious problem I've been stuck on. Suppose $G=H\rtimes K$, where $H$ is cyclic of order $n$, $K$ abelian with $\gcd(|K|,\varphi(n))=1$, $\varphi$ being the totient function. Why is $G$ ...
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7answers
451 views

Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group

Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group. This question is from the book 'Of Abstract Algebra' by Pinter. Now $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ containt 8 elements. ...
5
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3answers
111 views

Show that $\mathbb{Z}_{10}$ is generated by 2 and 5.

In the book 'Of Abstract Algebra' by Pinter the following question is asked: Show that $\mathbb{Z}_{10}$ is generated by $2$ and $5.\,$ Here, $,\mathbb{Z}_{10}\,$ is defined as the group of ...
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0answers
58 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
2
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2answers
158 views

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$? I have calculated that there are $24$ elements of order $10$ I know that in a cyclic subgroup of order ...
4
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2answers
399 views

Any group of order 12 must contain a normal Sylow subgroup

This is part of a question from Hungerford's section on Sylow theorems, which is to show that any group with order 12, 28, 56, or 200 has a normal Sylow subgroup. I am just trying the case for $|G| = ...
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0answers
77 views

Results on Hall $\pi$-subgroups - proof verification

I would appreciate verification of the following proofs. The problems are all closely related and the proofs are similar, so I think it's only appropriate to post them together (also, they are part ...
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1answer
1k views

A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$

This is problem 3 from Hungerford's section about the Sylow theorems. I have already read hints saying to use induction and that $p$-groups always have non-trivial centres, but I'm still confused. ...
2
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2answers
811 views

Possible Class equation for a group

Determine the possible class equation for a group of order 21? Until now I have found the following: $1+3+3+7+7$ $1+1+1+3+3+3+9$ $1+1+1+1+1+1+1+7+7$ $1+1+1+1+1+1+1+1+1+3+3+3+3$ $1+1+1+\cdots +1 ...
3
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1answer
223 views

Number of groups of order $p^n$, where $p$ is prime

for $n=1$, it is cyclic. so, the number is $1$ for $n=2$, it is Abelian. so, the number is $2$ for $n\geq 3$, I don't know. Can you recommend a book or link which can be helpful for understanding ...
1
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1answer
96 views

The group of rigid motions of an icosahedron.

Prove that group of rigid motions of icosahedron is isomorphic to $A_{5}$. Can you help me to prove this? What I have done is shown that the order of the group of rigid motions of icosahedron is ...
0
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1answer
54 views

Inner automorphisms of $S_3$

How do I prove that $S_3 \simeq \wp(S_3)$? So I must show that the group of inner automorphisms of $S_3$ is isomorphic to $S_3$. I haven't been given many examples on how to do these types of ...
0
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1answer
79 views

Prove that $N \setminus Z(G)\neq \langle e \rangle$.

Let $G$ be a group with $\operatorname{ord}(G) = p^n$, where $p$ is a prime number, and if $N \neq \langle e \rangle$ is a normal subgroup of $G$, prove that $N \setminus Z(G)\neq \langle e \rangle$.
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3answers
105 views

Examples of a map involving group actions

Okay, this is a trivial question but I need some non-trivial examples of a map involving group actions. What I mean: Let $G$ be a group acting on a set $A$. Let $G'$ be another group acting on ...
4
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0answers
65 views

Proving a group is cyclic [duplicate]

Let $G$ be a group of order $pq$, where $p,q$ are primes, $p < q$ and $q≢1$ (mod $p$); how do we prove that $G$ is cyclic ? (I have no idea)
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1answer
56 views

How to show that $S_4=A_3D_4$ and $S_4=A_3K$?

Ok, so I know how to do this by direct computation, but that seems like it will take a long time, and that is probably not how my professor wants this done. This is a practice question for the exam, ...
0
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2answers
77 views

For each pair in the list decide with proof if the groups are isomorphic

I have a question in my list of exercises and there is nothing in my lecture notes about it, and we havent done an example of anything similar. I missed a workshop due to illness so I fear I may have ...
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5answers
286 views

Prove that the order of any element in the additive group of integers modulo n is a divisor of n.

I am not opposed to struggle but I have been on and off of this problem for three days and need to present the proof tomorrow. I am thinking that because I know for any element in the additive group ...
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1answer
40 views

Sylow $q-$radical subgroup of a solvable group

Let ‎‎$G‎‎$ ‎be a‎ ‎finite ‎solvable ‎group ‎of ‎order ‎‎$p^2q^2‎‎$‎, where ‎$p>q‎$ ‎and ‎$‎q\nmid‎‎ ‎p-1‎$‎‎. ‎Let‎ ‎‎$G‎‎$ ‎has ‎the ‎following ‎presentation‎:‎ $‎‎‎\langle a‎ , ‎b‎ ,‎c \vert ...
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1answer
86 views

$S_n$ and its subgroups

Show that $A_n$ is unique in $S_n$ with index $2$. I'm trying to use Quotient Group and Lagrange's Theorem to approach this problem but I'm still clueless. Can anyone tell me how to do this problem? ...
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3answers
141 views

Finite group of two generators

My question is simple : Any finite group of two generators is cyclic, semidirect sum, or direct sum ?
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2answers
217 views

$A_5$ has no subgroup of order 15 and 20

Show that $A_5$ has no subgroup of order 15 and 20. I have been thinking about this problem for so much time but I'm still clueless. Can anyone tell me how to do this problem? Thanks. I ...
1
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1answer
46 views

Group algebras, Maschke's lemma and direct sums of matrix algebras

Let $G=\{g_1,g_2,\dots,g_n\}$ be an arbitrary finite group. We consider its representations over $\mathbb{C}$. There is Maschke's theorem which states that each representation of $G$ is a direct sum ...
2
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1answer
58 views

Looking for a direct proof for an identity in even-order groups

Let $G$ be a finite group of even order. I am interested in the following identity: $$\large{(x_1^2 x_2^2 \cdots x_n^2)^{\frac{|G|}{2}} = 1}$$ For arbitrary $x_i \in G$. I do know a certain proof, ...
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1answer
30 views

Abstract Algebra group question

If $G$ be a finite group of $l$ elements. Suppose that $a$ belongs to $G$, and $\mathrm{ord}(a)=k$,can $k>l$? I think $k$ can't be bigger than $l$, because $k$ should equal $l$.
3
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1answer
147 views

Why is conjugation by an odd permutation in $S_n$ not an inner automorphism on $A_n$?

I was reading about the outer automorphism group on wikipedia, and it mentions that conjugation by an odd permutation is an outer automorphism on the alternating group $A_n$. This suggests the ...
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2answers
47 views

Finding the order of a finite group

Let $x\in\mathbb{Z}/42$, and suppose that x has order $n\in\mathbb{Z^+}$. Without listing all of the subgroups of $\mathbb{Z}42$, determine all of the possible values that $n$ could be. I'm having a ...
0
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1answer
109 views

Properties of Permutations of a Set A

Let $A$ be a finite set, and $B$ a subset of $A$. Let $G$ be the subset of $S_A$ consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove that $G$ is a subgroup of ...
4
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3answers
167 views

Proving that two permutation groups are isomorphic

Here's the statement to prove: Let $n,m$ be two positive integers with $m≤n$. Prove that $S_m$ is isomorphic to a subgroup of $S_n$, where $S_n$ is the collection of all permutations of the set ...
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2answers
48 views

Prove that $b^2 \in \langle a \rangle$

Let $G$ be a group of order $8$. Assume that there exists $a \in G$ such that $|a|=4$ and that no element of $G$ has order $8$. Explain why $\langle a \rangle \lhd G$. Assume that $b \notin \langle ...
2
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1answer
108 views

Prove that this group of matrices has order $p^3$

Let $G$ be a group of upper triangular matrices $\in \mathcal{M}_3 (\mathbb{Z}_p)$ with ones on the diagonal. I've already proved that this group isn't abelian, but I don't know how to show that its ...
0
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1answer
147 views

There are only two types of groups of order $6.$

There are only two types of groups of order $6.$ Could anyone advise on how to prove a/m claim? Here is my attempt but I'm stuck: If $\exists g\in G$ such that $o(g) =6,$ then $G = \left ...
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1answer
44 views

Prove that $\langle a\rangle = G$, where $G$ a cyclic group of order 24, $a \in G$, $a^8 \ne e$, $a^{12} \ne e$.

Let $G$ a cyclic group, $|G| = 24$. Let $a\in G$, such that $a^8 \ne e$ and $a^{12}\ne e$. Prove that $\langle a \rangle = G$. So far i have found that $a^2,a^3,a^4,a^6 \ne e$. So to solve the ...
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1answer
74 views

Semi-direct product question

I have a semi-direct product that I feel must be nonabelian, but my thought process is telling me it is abelian. I have $G\cong Z_7\rtimes Z_4$ and I have indeed found nontrivial homomorphisms ...
0
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2answers
109 views

Find the subgroup $H$ of $A_4$ generated by $(123)$

I need to find the subgroup $H \leq A_4$ generated by $(123).$ I know that this subgroup will have order 1,2,3,4,6, or 12.
3
votes
1answer
88 views

Extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$

Given that $H^2(\mathbb{Z}_n,\mathbb{Z})=\mathbb{Z}_n$, it follows that up to equivalence there should be $n$ extensions of $\mathbb{Z}_n$ by $\mathbb{Z}$, one for each cohomology class. I'd like to ...
5
votes
1answer
169 views

Example of a Non-Abelian Group

I was hunting an example of a non-trivial finite group in which 1) All non-trivial normal subgroup are non-abelian. 2) There exists a nontrivial subnormal abelian subgroup. Is there any hope to ...
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2answers
75 views

Counting the Elements of a Finite Group [closed]

Let $G = \mathbb Z_6 \times \mathbb Z_4$, and find $[G:H]$ for: a) $H = \{0\} \times \mathbb Z_4$ b) $H = \langle 2\rangle \times \langle 2\rangle$
2
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1answer
96 views

If $G$ is a finite group, $H\vartriangleleft G$, $G/H$ is finite $p$-group and $H\subseteq Z\left(G\right)$, show that $[G,G]$ is $p$-group

If $G$ is a finite group, $H\vartriangleleft G$, $G/H$ is finite $p$-group and $H\subseteq Z\left(G\right)$, show that $[G,G]$ is $p$-group.
2
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0answers
95 views

Lift a group action from a quotient

Let $p$ be a rational prime and $H$ be a finite cyclic group of prime order $l$ prime to $p$, i.e. $(l,p) = 1$. Let $G$ be a finite abelian group of $p$-power order. If I can write an (abelian) group ...
0
votes
1answer
26 views

Why is the order of $X_{2n}$ is at most $6$ where $X_{2n}=\langle x,y\mid x^{n}=y^{2}=1,xy=yx^{2}\rangle$?

I am reading the book Abstract Algebra by Dummit and Foote. I am at the begining of the book, and I got to a section about generators and relations. The book gives the definition $$ X_{2n}:=\langle ...
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2answers
176 views

Sylow's Theorem Application. Prove $G$ is Abelian.

Assume that $|G|=5^27^2$. Determine the possibilities for $n_5, n_7$ and determine what can be concluded in each case about the $5$-Sylow subgroups and the $7$-Sylow subgroups and prove that $G$ is ...
0
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1answer
28 views

Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion?

I am reading the book Abstract Algebra by Dummit and Foote. In the section about the group $D_{2n}$ (of order $2n$) the authors claim that knowing where two adjacent vertices move to, completely ...
0
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0answers
72 views

Showing that $ U(2^n) $ is not a cyclic group for $ n \geq 3 $ [duplicate]

Could anyone please explain to me why $ U(2^n) $ is not a cyclic group for $ n \geq 3 $? I need help on this because I have an algebra exam tomorrow. Thanks!