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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5
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1answer
336 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 ...
7
votes
2answers
247 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 ...
1
vote
1answer
108 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. ...
3
votes
1answer
90 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 ...
5
votes
2answers
322 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 $ ...
3
votes
1answer
209 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\}, ...
1
vote
3answers
145 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 ...
3
votes
1answer
417 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 ...
2
votes
1answer
99 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 ...
4
votes
1answer
324 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 ...
7
votes
1answer
751 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 ...
5
votes
4answers
966 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 ...
5
votes
1answer
117 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 ...
4
votes
3answers
311 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 ...
2
votes
0answers
123 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 ...
1
vote
2answers
256 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 ...
2
votes
1answer
313 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 ...
1
vote
1answer
313 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 ...
1
vote
1answer
241 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 ...
1
vote
2answers
138 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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vote
0answers
72 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 ...
5
votes
3answers
959 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...
0
votes
1answer
226 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 ...
0
votes
0answers
90 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 ...
2
votes
1answer
220 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 ...
2
votes
1answer
429 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 ...
1
vote
1answer
85 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 ...
2
votes
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.
2
votes
2answers
158 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 ...
1
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1answer
148 views

What are the generators for $\mathbb{Z}_p^*$ with p a safe prime?

lets consider $\mathbb{Z}_p^*$ with $p = 2 \cdot q + 1$ a safe prime ($p$ and $q$ have to be prime). Then $\varphi\left(p\right) = 2 \cdot q$ is the order of $\mathbb{Z}_p^*$, and ...
2
votes
2answers
209 views

An Equivalence class in a group of permutation

We know that every equivalence relation, induced by a partition on a set for example $X$ , make some equivalence classes. Now, if a group $G$ acts on $X$ then the associated equivalence classes are ...
4
votes
2answers
645 views

Nilpotent groups are solvable

I know this should be obvious but somehow I can't seem to figure it out and it annoys me! My definition of nilpotent groups is the following: A group $G$ is nilpotent if every subgroup of $G$ is ...
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3answers
1k views

How to find the order of elliptic curve over finite field extension

I want to find the order of elliptic curve over the finite field $\mathbb{F}_{5^2}$, where $E(\mathbb{F}_{5^2}):y^2=x^3+10x+17$. I am using the method illustrated by John J. McGee in his thesis ...
8
votes
1answer
278 views

Generators for $S_n$

This problem is from Dixon's book telling that; if $x$ is any nontrivial element of the symmetric group $S_n$ and $n≠4$, then there exists an element $y\in S_n $ such that $S_n=\langle x,y\rangle$. ...
2
votes
1answer
394 views

List Proper Primitive Groups of a certain degree by GAP

As J.D.Dixon noted in his great books, there are just 5 proper primitive groups of degree 8, $P(8)=5$. I wanted to examine it with GAP, so wrote the following small program: > G:=[];; > for k in ...
5
votes
1answer
1k views

An abelian group of order 100

The first part of the problem asks you to prove that an abelian group $G$ with order $100$ must contain an element of order $10$. For this part, I use Sylow theorem to list possiblities for $H$ and ...
3
votes
1answer
160 views

Find the subgroups of index two of this finite semi-direct product

This older stackoverflow question may be helpful in answering the question that I ask below, although I could not work it out. For $n\geq 1$, let $X=\lbrace 1,2, \ldots ,n \rbrace$, $Y=X \cup (-X)$ ...
2
votes
1answer
362 views

Why is this subgroup normal?

Let $G$ be a group of odd order and $H$ a subgroup of index 5, then $H$ is normal. How can I prove this? (there is an hint: use the fact that $S_5$ has no element of order $15$) I took the usual ...
6
votes
1answer
1k views

Converse of Lagrange's theorem for abelian groups

I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...
7
votes
1answer
267 views

Reference for classification of small groups

There are various online resources for the classification of groups of small order, such as this one or that one. Is there any nice reference in the literature which contains such a classification ...
5
votes
4answers
321 views

About the Order of Groups

Is there any theorem that states the all the finite groups of order n are the same? or some sort of theorem that refers to the order of two finite groups? If anyone can post a reference to this topic ...
3
votes
1answer
74 views

Why are generators of $Z^{*}_p, p=c \cdot 2^k + 1$ so small?

I was implementing NTT for long integer multiplication and thus searched for generators of several $Z^{*}_p$ groups where $p=c\cdot 2^k + 1$. I used the algorithm described in Wikipedia which uses ...
1
vote
1answer
82 views

Infinitely many nilpotent elements in $\mathbb{C}[G]$

Suppose $G$ is a finite group and $F$ is a field such that $\mathrm{char}\;F$ doesn't divide $|G|$. Suppose that $F$ is algebraically closed and $G$ is not abelian. How can I prove that $F[G]$ has ...
4
votes
1answer
216 views

Primitive root modulo $p$

I am currently trying to find a primitive element of the multiplicative group of field $GF(p)$. Since the numbers are relatively small, I know the factorization of $$\phi(p)=p-1 = {p_1}^{k_1} ...
3
votes
1answer
135 views

Why is it that any two distinct subgroups of G of order p (prime) intersect in 1?

Why is it that any two distinct subgroups of G of order p (prime) intersect in 1? It says so here on page 29. But why is it?
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0answers
88 views

Is it true that if $g^{-1} = g$, then $g$ is only conjugate to itself in a finite group $G$?

I have a finite group G, where $Z(G)=1$, and I have an element $g\neq 1$ where $g^{-1} = g$ and $gg=1$. I want to say that $g$ is then only conjugate to itself so that I have a contradiction ($Z(G)$ ...
3
votes
1answer
151 views

Is my textbook wrong about this corollary of Sylow's theorem?

Is my textbook wrong about this corollary of Sylow's theorem? Let $G$ be a finite group and $p$ a prime that divides $|G|$. Let $n_p$ be the number of Sylow $p$-subgroups of $G$. Then $n_p = 1$ if ...
2
votes
1answer
114 views

Group acting transitively on subset and what must hold for that subset

I've been struggling with this problem for some time now so any help would be greatly appreciated. I have a finite group $G$ and a subset $A$ of $G$. I am told that $G$ acts transitively on $A$ and ...
1
vote
1answer
703 views

Prove that every p-group is nilpotent.

Let $G$ be a p-group $|G| = p^n$. Then G is nilpotent in the following sense: Let $C^1(G) = [G,G] = G'$ And $C^n(G) = [G,C^{n-1}]$. G is nilpotent iff there is a $n_0$ such that $C^n = \{e\}$. What ...
0
votes
1answer
42 views

Finite group - binary

Prove that $g(\alpha)$=0 if and only if $g'(\alpha)$=0 $g(t)=t^{11}+t^{10}+t^6+t^5+t^4+t^2+1$ $g'(t)=t^{11}+t^9+t^7+t^6+t^5+t+1$ where $\alpha \in F[t]$. We are working in standard binary space. ...