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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$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
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3answers
207 views

Proof that every element of A_5 is an involution or a product of two involutions?

It can be verified with brute force that the alternating group on 5 elements ($A_5$) has the property that every member is either an involution or can be written as the product of two involutions. Is ...
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1k views

On automorphisms group of direct product of two groups

Let $G$ and $H$ be two finite groups. We know if $\alpha\in \mathrm{Aut}(G)$ and $\beta\in \mathrm{Aut}(H)$, then $\alpha\times\beta\in \mathrm{Aut}(G\times H)$. Hence, $| \mathrm{Aut}(G\times ...
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0answers
36 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
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1answer
74 views

Is there any permutation $x≠1$ leaving at least $n-2k$ letters fixed at this group?

This question has an answer which I am noting both here. Q: Suppose that $G$ is permutation group of degree $n$. If for an integer $k$ where $4≤2k≤n$ we have $|G|≥(n-k)!k$ then $G$ contains a ...
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1answer
165 views

On the $PSL(2, p)$, $p$ a Mersenne prime

We know $|PSL(2,p)|=p(p+1)(p-1)/2$. Let $p$ be Mersenne prime (that is $p+1=2^{n}$) and $r$ be prime divisor of $(p-1)/2$. My question: What is the number of Sylow $r$-subgroups of $PSL(2,p)$?
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2answers
151 views

Question about non cyclic group of order $n$ with Euler function

Let $n >1$ such that $(\mathbb{Z}/n \mathbb{Z})^\times $ is not cyclic. Then show that for each $a \in \mathbb Z$ with $\gcd(a,n)=1$ we have $a^{\frac{\varphi (n)}{2}} \equiv 1 \mod n$. (Here ...
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1answer
185 views

If $G$ is a transitive permutation group then $\mathrm{fix}(G_\alpha)$ is a block

I am new here and don't know much about Latex so, I attach my question from Permutation Groups by J. Dixon. I hope to get a help for it: 1.6.5 Let $G$ be a transitive subgroup of ...
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1answer
189 views

For which $n$, $G$ is abelian?

My question is: For Which natural numbers $n$, a finite group $G$ of order $n$ is an abelian group? Obviouslyو for $n≤4$ and when $n$ is a prime number, we have $G$ is abelian. Can we consider ...
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329 views

Proving $H$ is normal in $G$. [duplicate]

Possible Duplicate: Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$. I have to solve the following problem. It's an ...
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1answer
341 views

Does $abab=baba$ imply commutativity in a Group of uneven order?

Suppose $(G,\cdot)$ is a finite group of uneven order such that $abab=baba$ for any $a,b\in G$. Does this mean that $G$ is commutative?
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1answer
387 views

Determining all Sylow $p$-subgroups of $S_n$ up to isomorphism?

I'm trying to understand a classification of all Sylow $p$ subgroups of $S_n$. Let $Z_p$ be the subgroup of $S_p$ generated by $(12\cdots p)$. Then $Z_p\wr Z_p$ has order $p^p\cdot p=p^{p+1}$, and ...
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1answer
103 views

Is every finite group an extension?

I would like to know if every finite group is an extension of some group by another. Thanks
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2answers
229 views

$p$-Sylow subgroups of a group of order $5^3\cdot 29^2$

I need to calculate the $p$-Sylow subgroups of a Galois group with order $5^3 \cdot 29^2$, i.e. $|\mathrm{Gal}(K/F)|=5^3 \cdot 29^2$. I've already established that there is only one 29-Sylow-subgroup ...
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0answers
75 views

Concerning the point stabilizing group and coset stabilizing group.

I would like to know more about the point stabilizer group and the coset stabilizer group, like the definitions, why they are used in group theory, who developed them and there importance.
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5answers
213 views

Are there methods to well order a finite group in a meaningful way?

Can some finite groups be well ordered in a "meaningful" way? I mean, it is clear that we can trivially find a bijection between $\{1,...,n\}$ and a finite group $G$ with $n$ elements, but I am ...
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3answers
222 views

Rotation group, altitude

Could someone give me a rigorous proof that the group of rotations each element of which is a composition of rotations around the altitudes of a tetrahedron that transform the tetrahedron into itself ...
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2answers
219 views

Exhibiting a special subgroup whose involutions are all conjugate to a given involution?

I'm trying to work through a sketch proof attributed to Walter Feit on characterizing $S_5$. Suppose $G$ is a finite group with exactly two conjugacy classes of involutions, with $u_1$ and $u_2$ ...
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3answers
133 views

Computing the length of a finite group

Can someone suggest a GAP or MAGMA command (or code) to obtain the length $l(G)$ of a finite group $G$, i.e. the maximum length of a strictly descending chain of subgroups in $G$? Thanks in advance.
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2answers
938 views

A group of order $p^2q$ will be abelian

This problem is not homework but, I was stuck to it when I reviewed the Sylow theorems and problems. I am really interested of finding a test in which we can examine whether a finite group of certain ...
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1answer
116 views

Center of SO(V,q)

Let $V$ be finite dimensional vector spaces and $q$ is quadratic form. I'm looking for $Z(SO(V,q))$. where $SO(V,q)$ is special orthogonal group. If $\operatorname{dim} V$ is odd then ...
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1answer
384 views

Any nonabelian group of order $6$ is isomorphic to $S_3$?

I've read a proof at the end of this document that any nonabelian group of order $6$ is isomorphic to $S_3$, but it feels clunky to me. I want to try the following instead: Let $G$ be a nonabelian ...
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2answers
282 views

Estimates on conjugacy classes of a finite group.

In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32. Theorem: Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
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1answer
110 views

For which numbers there is only one simple group of that order?

There is only one simple group of orders: 3, 60, and 360 respectivley. Are there other groups of this kind? What general characteritics do they share? From pure curiosity did this question arise. ...
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1answer
95 views

Is there any finite non trivial Group with this property?

I was asked to have a look at a problem: There is no a finite non-trivial group $G$ that all its non-trivial elements can be commuted with exactly half elements of group . For the first step, I ...
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2answers
358 views

If $G$ is finite and abelian, then every subgroup of $G$ is characteristic if and only if $G$ is cyclic

Suppose $G$ is finite and abelian. Show that every subgroup of $G$ is characteristic if and only if $G$ is cyclic. I have the 'if' part so far: If $G$ is cyclic, then $G = \langle g \rangle $ ...
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1answer
225 views

Inducing a representation from a subgroup.

Find all the irreducible representations of the group given by: $<x,y,z|x^2=y^2=(xy)^2=z^6=1,zxz^{-1}=y, zyz^{-1}=xy>$. I have 8 conjugacy classes: $\{1\}, \{z^3\}, \{x,y,xy\}, ...
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3answers
153 views

Order of a subgroup of a finite cyclic group

Let $G$ be a cyclic subgroup of order $n$, generated by say $a\in G$ where the identity of $G$ is labelled $e$. Let $H$ be the cyclic subgroup of $G$ generated by some $a^{m}\in G$. Then I want to ...
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1answer
484 views

Find all non-abelian groups of order 105.

Find all non-abelian groups of order 105. My attempt: $105=3.5.7$. Consider the $3$-factorization, as $5, 7, 35 \not\equiv 1\mod 3$ we have that the sylow $3$-subgroup is normal. Consider now the ...
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1answer
104 views

How to bound the order of a finite group under the following hypotheses?

In the book Character Theory Of Finite Groups by I.Martin Issacs as exercise 2.14 Let $G$ be a finite group with commutator subgroup $G'$. Let $H \subset G' \cap Z(G)$ be cyclic of order n and ...
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1answer
351 views

A representation is semisimple if its restriction to a subgroup of index prime to Char(F) is semisimple

Let $G$ be a finite group and $H$ a subgroup whose index is prime to $p$. Suppose $V$ is a finite-dimensional representation of $G$ over $\mathbb{F}_p$ whose restriction to $H$ is semisimple. Prove ...
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1answer
810 views

Non-abelian $p$-group; abelian subgroups of index $p$

I'm trying to prove the following problem: (a) Let $G$ be a non-abelian $p$-group with an abelian subgroup of index $p$. Then the number of abelian subgroups of $G$ of index $p$ is either $1$ or ...
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4answers
1k views

About alternating group $A_4$

This is a simple exercise telling that $A_4$ cannot have a subgroup of order $6$. Here in my way: Obviously, for any group $G$ and a subgroup $H$ of it with index $2$; we have $∀$$ g\in G$ ,$g^2\in ...
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1answer
120 views

Is there an algorithm to determine whether rational matrices generate a finite group?

This is inspired by this question. Given finitely many invertible rational $n\times n$ matrices $A_{1},\ldots, A_{k}\in\operatorname{GL}(n,\mathbb{Q})$, is there an algorithm (a practical one) to ...
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3answers
319 views

Show a certain group is contained in a Sylow p-group.

Statement: Let G be a group and p a prime that divides $|G|$. Prove that if $K\le G$ such that $|K|$ is a power of p, K is contained in at least one Sylow p-group. I just started studying Sylow ...
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0answers
130 views

Structure of elementary abelian group by a cyclic group of coprime order

My mind is a little foggy right now, so I would like to ask for some help on this (or an appropriate reference). Suppose $A$ is an elementary abelian $p$-group and $B$ is a cyclic group of order ...
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2answers
308 views

Powers of elements and subgroups

Let $(G,\circ)$ be a group and $N\subseteq G$ a normal subgroup of order $n<\infty$ and let $g\in G$. Is the element $g^n$ in $N$? Given a subgroup $H\subseteq G$ of order $n$, is element $g^n$ in ...
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1answer
358 views

Find an inverse element for an element in this Group

We know that if $\Omega$ be set of all 1-dimension subspaces of $V=V_{2}(q)$ which $V$ is a vector space on a finite field $GF(q)$ and so $|V|=q^{2}$ then, group $PGL_2(q)$ acts on $\Omega$. Also, it ...
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1answer
342 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...
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1answer
254 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
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2answers
145 views

Frobenius group as semidirect product of finite group with a regular group of automorphisms.

Let $G$ be a finite group. We say a non-trivial group of automorphism $A$ on $G$ is regular, if each non-trivial automorphism of $A$ is regular, i.e. fixes only the identity. It is remarked in ...
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76 views

What could the meaning of “invariant of $G$” be?

In a context, the author wrote that for a finite permutation group $G$ acting on a set $\Omega$ of degree $n$, the list of subdegrees is an invariant of the group. What could the meaning of ...
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3answers
1k views

Question about normal subgroup and relatively prime index

Suppose $G$ is finite, $K$ is a normal subgroup in $G$, $H$ is a subgroup of $G$, and $|K|$ is relatively prime to $[G:H]$. Show that $K$ is a subgroup of $H$. I don't know where to begin...
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1answer
234 views

Are actions in the $3\times 3\times 3$ rubik cube a group?

Are actions in the $3\times 3\times 3$ rubik cube a group? You can see here Rubik's Cube Not a Group? that $4\times 4\times 4$ rubik cubes or higher arent groups. But what about $3\times 3\times ...
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1answer
114 views

Conjugates of Transpositions

I've been asked the following. Assume $S_n$ is generated by the adjacent transpositions $(1,2),(2,3),...,(n-1,n)$ Let $\sigma \in S_n$. Calculate the conjugate of the transposition $(a,b)$ by ...
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1answer
248 views

proof that finite group of rotations of plane is cyclic

How can we prove that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic, using the following hint? 'You may assume that given any non-empty finite set E in the ...
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1answer
515 views

All groups of order 175 are abelian?

Question. All groups of order 175 are abelian? I can show that there exists only one Sylow 5-subgroup of order 25, call it $H$, and one Sylow 7-subgroup of order 7, denote $K$. I know that $K$ is ...
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1answer
87 views

Meaning of Strong Primitivity

As J.D.Dixon noted in his great book; Permutations Group, we can speak about Strong Primitivity of a group acting on a set $\Omega$ by means of orbital graphs. The way he paved employes digraphs prove ...
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1answer
73 views

Is this true that $p^2\big||Aut(G)|$?

Is this statement true that for a finite and non abelian $p$-group $G$; $p^2\big||Aut(G)|$? I just found $Q_8$ fulfillment of this claim.
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2answers
163 views

equivariant hyperplane sections

Suppose you are given a smooth algebraic variety $X$ inside a projective space $\mathbb{P}$ and that there is a linear action of a finite cyclic group $G$ on $\mathbb{P}$ which restricts to an action ...