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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149 views

Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
2
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2answers
92 views

$N$ is a normal subgroup of $G$(Herstein)

Let $N$ be a normal subgroup of $G$. Suppose that the order of $N$ and index of $N$ in $G$ are co-prime. Prove that $\{x|x\in G ;x^{|N|}=e\}$ is a normal subgroup of $N$.
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1answer
57 views

Group generated by normal subgroup and one more element.

Suppose I have a normal subgroup $N$ in $G$, and suppose there exists an element of order $p$ in $G/N$; i.e., some coset $gN \in G/N$ has order $p$. I'm trying to find another subgroup in $G$ that ...
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1answer
46 views

Confusion in Burnside's proof that any $2$-generated group of exponent $3$ is finite?

I'm reading a proof of Burnside's theorem that groups of exponent $3$ are finite, but have some problems. Let $G=\langle x,y:z^3=1\rangle$ be a group generated by $x$ an $y$ with exponent $3$. Let ...
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2answers
109 views

Question about Abelian group proof

I prove that if $G$ is Abelian group so if $a,b\in G$ has a finite order so $ab$ has a finite order to.. (Maybe later I'll upload here my proof to see of she is correct....) Now, I have to show that ...
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1answer
51 views

What is the order of $e^{\large \frac{4\pi i}{5}}$ in the circle group $U_{20}$?

We talk about the Circle group. What is the order of $e^{\large \frac{4\pi i}{5}}$? The power is $\frac{4\pi i}{5}$ if it's not clear... Thank you!
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2answers
163 views

show that $S_7$ has no subgroup of order 11

how to show that $S_7$ has no sub group of order 11 Is it a proof by contradiction, whereby, let H be the subgroup of order 11, assume H is a subset of subset of $S_7$, then by lagrange's theorem we ...
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1answer
59 views

Subgroup of a nilpotent group

Let $G$ be nilpotent and $H \le G$. Let $P_1,P_2,\ldots,P_k$ be the Sylow subgroups of $H$. Is it true that $H = P_1 P_2 \cdots P_k$? I know that when $G$ is nilpotent, it is the direct product of ...
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2answers
114 views

What is $\mathbb{R}/\mathbb{Z}$ isomorphic to?

Intuitively, I see how is related to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$. I tired to use the first Isomorphism theory where it states that the image of φ is isomorphic to the quotient group G ...
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3answers
277 views

Symmetries of combinatorial structures.

Studying the automorphism groups of graphs/finite geometries/designs has been quite useful and important for both group theory and combinatorics. I know of the following books which cover the ideas ...
3
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1answer
137 views

A Sylow $p$-subgroup can be a subset of union of the rest of the Sylow $p$-subgroups?

Let G be a finite group and assume that number of the Sylow $p$-subgroups is more than one. My question is this: "Can a Sylow $p$-subgroup be a subset of the union of the rest of the Sylow ...
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1answer
80 views

Bijection map from a set of subgroup to another set of subgroup under some condition.

Let $G$ be a finite group. Let $N \lhd G$ and $U \leq G$ such that $G = NU$. Then there exists a bijection, preserving inclusion, from the set of subgroups $X$ satisfying $U ≤ X ≤ G$ to the set of ...
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1answer
66 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
81 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 ...
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0answers
286 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
94 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 ...
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3answers
215 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
73 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
42 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
469 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. ...
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3answers
125 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
66 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 $g \in H$?
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2answers
178 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 ...
5
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2answers
546 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| = ...
2
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0answers
123 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
2k 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. ...
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3answers
1k 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
252 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 ...
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1answer
117 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
62 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 ...
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2answers
96 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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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 ...
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0answers
70 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)
3
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1answer
59 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, ...
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2answers
89 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
434 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
44 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
87 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
175 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
294 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
51 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, ...
0
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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 $l$.
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1answer
161 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
48 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
129 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
222 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
49 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
votes
1answer
119 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
215 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 ...