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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85 views

Structure of $C(F_5)$ from Rational Points on Elliptic Curves

In the book Rational Points on Elliptic Curves by Silverman/Tate one examines the elliptic curve $y^2 = x^3 + x + 1$ over $F_5$. One can then easily determine the group $$ C(F_5) = \lbrace ...
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1answer
291 views

A group of order 561 is cyclic.

Prove that any group of order 561 is cyclic.
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1answer
130 views

Existence of a solution for an equation in a permutation group

Here is a concrete example, but I'm looking for methods in general : Let $S_{13}$ be the permutation group. Let $i : S_2 \times S_3 \times S_4 \times S_4 \to S_{13}$ be the canonical injection. Let ...
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1answer
45 views

Any group of order 2013 has a subgroup for each divisor of 2013

I want to prove that for any group of order $2013$ and any divisor of $2013$ there is some subgroup of that order. $2013 = 11 \times3 \times61$ so we have a normal $11$-Sylow subgroup. Can ...
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2answers
69 views

$H_1,H_2$ distinct subgroups of $G$ each of order 2 , $H$ be smallest containing both , what is order of $H$

Let $H_1,H_2$ be two distinct subgroups of finite group $G$ each of order $2$. Let $H$ be the smallest subgroup containing $H_1$ and $H_2$ . Then is it necessary that order of $H$ is amongst ...
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2answers
98 views

Let $A$ be a finitely generated abelian group. Show that $\operatorname{Hom}(A,Z)$ is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
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2answers
113 views

finding all composition series of ${\rm sym} \ (4)$

How can I find all composition series of ${\rm sym}\ (4)$ ? I think first I have to find all maximal normal subgroups. But how ? Thanks.
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1answer
46 views

Finding the group generated by 2 given 3 * 3 binary matrices

Having trouble completing this exercise. I posted a few questions on subgroups generated by subsets of a group. But am still at odds on how to solve a problem of this type. The orders of the first ...
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1answer
61 views

A weaker version of Sylow's theorems

Prove that the number of normal subgroups of order $p^s$ of a finite $p$-group $G$ is congruent to $1$ mod $p$. I know this result is a weaker version of Sylow's theorems. But without using ...
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2answers
140 views

Groups of order $56$

Let $G$ be a group of order $56$. (We do NOT assume the Sylow-$7$ subgroup to be normal.) Then either the Sylow-$2$ subgroup is normal or the Sylow-$7$ subgroup is normal. How to prove? My idea: ...
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2answers
568 views

What are a few examples of noncyclic finite groups?

I just want to make sure that every group $\mathbb{Z}_n$ for any $n$ is cyclic. Further, every group of prime order is cyclic because it is isomorphic to $\mathbb{Z}_p$. I think I have a handle on ...
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1answer
84 views

Schur's Theorem about the derived subgroup

(Schur) Suppose Z(G) is of finite index in G, then the derived subgroup of G is finite. We know Schur's lemma that says: Let |G:Z(G)|=m. Then the map g to g^m is a homomorphism from G into ...
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1answer
50 views

Smallest group containing two given elements

Given two elements of a finite group a, b Ԑ G, what is the smallest group containing both these elements? Is it {axby| x = 0, 1,...., m & y = 0, 1,...., n} ? OR Is it the Union, {axby| x = ...
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1answer
253 views

Prove that if |G|=132 then G cannot be simple

Okay so I have done this but I would like a heads up if it is enough to prove it. $132=2\cdot 2\cdot 3\cdot11=2^2\cdot3\cdot11$ Let us assume that G is simple. Then from Sylow's theorem, we can say ...
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2answers
61 views

$\cap_{g\in G}A^{g}$ contains a Sylow $p$-subgroup of $G$

Let $G$ be a finite group, and $A$ a subgroup of $G$. If the index $\left|G:A\right|$ is less than some prime divisor $p$ of the order of $G$, prove that $\cap_{g\in G}A^{g}$ contains a Sylow ...
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2answers
640 views

Prove that in any group an element and its inverse have same order. [closed]

Prove that an element and its inverse have same order in any group.
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64 views

Two questions about order of subgroups

We talking about $\mathbb{Z}_{66}\times \mathbb{Z}_{35}$. $\gcd(66,35)=1 \Rightarrow\;\mathbb{Z}_{66}\times \mathbb{Z}_{35}\;$ is cyclic. A. I need to find a subgroup with order 210, and tell how ...
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1answer
57 views

Prove something with finite groups and homomorphism…

$H,G$ are finite groups, and $\gcd(|H|,|G|)=1$. I need to prove that if $\varphi:G\to H$ is homomorphism it must be the Trivial Homomorphism. My try: I assume that $\varphi:G\to H$ is homomorphism. ...
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1answer
80 views

Prove that $\varphi$ is automorphism

$G$ is commutative group. $|G|=n$. $m\in \mathbb{N}$ and $\gcd(m,n)=1$. I need to prove that $\varphi :G\to G$, $\varphi(x)=x^m$ is automorphism of G. My try: I assume that $a\in \ker(G)$, so $a\in ...
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0answers
162 views

Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is ...
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1answer
134 views

Representation theory and characters

I have been studying representation theory for 6 months now. I came across the following question in a graduate course example sheet. Let $\chi$ be the character of a representation $\rho$ of ...
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2answers
63 views

What are the subgroups of $\mathbb{Z}_{5}^\ast \times \mathbb{Z}_{4}^\ast$? I think I count $5$ of them.

What are the subgroups of $\mathbb{Z}_{5}^\ast \times \mathbb{Z}_{4}^\ast$? I think I count $5$ of them I think? Thank you for anyone that puts the time in to help!
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59 views

Isomorphic subgroups in $\mathcal{Z}_4 \times \mathcal{U}_4$

Consider the group $ \mathcal Z_4\times\mathcal U_4$, where $\mathcal U_4$ is the group of units modulo $4$. Let $h=\{([2]_4,[3]_4)\}$ and $K=\{([2]_4,[1]_4)\}$. (Remember that the product ...
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2answers
118 views

Let $n=2m$, where $m$ is odd. How many elements of order $2$ does the group $D_n/Z(D_n)$ have?

Let $n=2m$, where $m$ is odd. How many elements of order $2$ does the group $D_n/Z(D_n)$ have? I don't how to begin this proof. All I have so far is that Dn/Z(Dn) should have one element of order ...
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1answer
115 views

Action of Modular Group on Finite Field

I need to define an action of Modular Group on Galois Field(7). Someone can define me this programe works for it? In this program I could not follow that why is he taking ...
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3answers
2k views

Prove that a finite abelian group is simple if and only if its order is prime.

So I'm having trouble with this problem. I know that the definition of a simple group means that the group has no nontrivial subgroups. I know that this can be proven somehow with the help of the ...
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1answer
49 views

Question about groups and subgroups

$G$ is finite group and $H,K$ are subgroups of $G$. How I prove that: $$K\subset H \Rightarrow [G:H]\cdot [H:K]=[G:K]$$ Id like to get hints for the proof. Thank you!
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3answers
44 views

Question about subgroups and Lagrange thorem

If $G$ is a group and $H,K$ are subgroups of $G$, how do I prove with Lagrange Theorem, that $$\gcd(|H|,|K|)=1⇒H∩K={e}$$ Thank you!
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1answer
97 views

Do coset representatives of a normal group form a subgroup of G?

Let G be a FINITE group and N is a normal subgroup of G . Let K be a set of representatives of the cosets of N . Is K a subgroup of G ? is there any specific condition for them to form a subgroup of ...
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1answer
38 views

Generated group of 2 element of order 2

Let $G$ be a finite group that is generated by $\alpha,\beta\in{G}$ of order 2, such that their product isn't of order 2. Show that $G$ is isomorphic to $D_n$ for some n.
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273 views

How many Groups there are on a finite set?

Let say cardinality of set S is $n=|S|$. We know that there are $n^{n^2}$ all binary operations on that set. To find out how many groups can be created by this set and by those operations, we need not ...
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1answer
82 views

A question about $PSL(2,8)$

Can anybody tell me how to construct the character table of $PSL(2,8)$? I need a specific method.
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75 views

$F$ a finite field of $p^n$ elements. Suppose $F^\times=\langle x \rangle$. Then $\phi(x)=x^{p^r}$ for automorphisms

Let $p$ be a prime and $F$ a finite field of $p^n$ elements. Suppose $F^\times=\langle x \rangle$. Let $\phi$ be an automorhpism of $F$. Then prove that $\phi(x)=x^{p^r}$ for some integer $r$. How to ...
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3answers
55 views

How would you find the order of an element in this case?

Let $G$ be a group with $|G| = p^2q$, where $p$ and $q$ are distinct primes. Let $N$ be a normal subgroup of $G$ with $|N| = p$. Let $x$ be an element of $G$ with $\text{ord}(x) = q$. What is the ...
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1answer
48 views

Abelian Groups properties in finite groups

How we can see if a group is abelain then every element has its own inverse? Thanks
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94 views

Existence of a subgroup with order 3 in a group with order 6

Let $G$ be a group of order 6. Why does $G$ has a subgroup of order 3 even if $G$ isn't cyclic? I've tried using to use negation and assume all elements in $G$ have an order of 2 or 1 but I ...
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1answer
30 views

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$.

Let $F$ be a finite field and $\tau$ an element of $F$. Prove that there exists $a,b\in F$ such that $\tau=a^2+b^2$. It suffices to prove for the case $F=\mathbb{Z}_p$. How to prove?
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1answer
47 views

Group order and left cosets

Let $G=S_3 \times \mathbb{Z}_4$ be a group and $H= \langle(1 2 3)\rangle \times \langle2\rangle$ and $K=\langle(1)\rangle \times \langle2\rangle$ be subgroups. Find the order of $G$, $H$ and $K$. ...
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1answer
118 views

Normal subgroups of group with order $p^2 $

Let $G$ be a group of order $p^2$ for a prime $p$. Show that $(a)$ There exists a subgroup $N$ of order $p$ which is normal. $(b)$ Any group $K$ of prime order is cyclic. $(c)$ Groups ...
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1answer
137 views

If every quotient by normal subgroup is abelian, then the irreducible representations are injective

The following is Problem 6 of January 2006 algebra qualifying exam from University of Maryland. See here for the problems. Let $G$ be a finite group. Suppose that for each normal subgroup $K\neq ...
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0answers
46 views

For what numbers $n$ every group of order $n$ is abelian? [duplicate]

What are the numbers $n$ such that every group of order $n$ is abelian? For every prime $p$, every group of order $p$ or $p^2$ is abelian. If there is a prime $p$ such that $p^3\mid n$, then I ...
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Groups of prime squared order

I have done part $a$ and $b$, stuck on $c$. I don't know what to do here. I know that since $G$ has order $p^2$, any element must have order $1, p$ or $p^2$ so any generator must have one of those ...
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2answers
135 views

Intuitive idea on generators of $S_4$

What is the way to convince myself that $\left\langle(1,2),\ (1,2,3,4)\right\rangle=S_4$ but $\left\langle(1,3),\ (1,2,3,4)\right\rangle\ne S_4$? Let $\sigma$ be any transposition and $\tau$ be ...
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4answers
103 views

Is there any nice characterization of the extension of $G/[G,G]$ by $[G,G]$ that equals $G$?

In this question, I asked whether a finite group $G$ could always be expressed as the semidirect product of its commutator subgroup $[G,G]$ and the abelian quotient group $G/[G,G]$. The answer is no. ...
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2answers
89 views

Prove that the lattice graph of $D_{16}$ is not planar

How do we prove that the lattice graph of $D_{16}$ is non-planar? I wanted to prove it using Kuratwoski's Theorem but was unable to do it. And to add one more question, are there any interesting ...
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4answers
95 views

Subgroup of order $n-1$ of a group of order $n$

Here is question 2.1.5 from Dummit and Foote : Prove that $G$ cannot have a subgroup $H$ with $|H| = n-1$, where $n = |G| > 2$. How can one show this without using Lagrange's theorem (which is in ...
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3answers
277 views

Will $G$ have an element $ab$ of order lcm$(|a|, |b|)$ if $|a|$ and $|b|$ are not necessarily relatively prime?

If $G$ is a finite abelian group, $a, b\in G$ such that $|a|=m>1, |b|=n>1$ where $m, n$ are not necessarily relatively prime, then prove or disprove that $G$ has an element $ab$ of order ...
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2answers
89 views

$G$ a finite $p-$group, $H < G$ (i.e, $H$ a subgroup of $G$, but $H \neq G$), prove that $H \neq N_G(H)$ [duplicate]

I have one problem, that I think it's pretty hard to solve. The problem reads: Let $G$ be a finite $p-$group, and $H < G$ (i.e $H \le G$, and $H \neq G$). Prove that $H \neq N_G(H)$. Here are ...
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2answers
196 views

Is $G$ always a semidirect product of $[G,G]$ and $G/[G,G]$?

If $G$ is a finite group, it is not true in general that $G$ is the semidirect product of a normal subgroup $N$ and the quotient group $G/N$. It is also not true in general that there is a subgroup ...
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3answers
148 views

Trouble understanding this proof about cyclic groups of order $p^n$

This is a theorem / proof from Rotman, and I am having a little trouble following it. I've reproduced the theorem and proof (proof is not verbatim) below. Theorem: Let $p$ be a prime. A group ...