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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Lluís Puig notes in Frobenius systems.

I have been reading some articles that makes mention about certain investigator called Lluís Puig and his theorems and some notes made by him about Frobenius Systems, but in the references of each ...
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26 views

If $G$ is cyclic of order $2$ show $|H^1(G,\mathbb{Z})|=2$.

Let $G$ be the cyclic group of order $2$ acting by inversion on $\mathbb{Z}$. Show $|H^1(G,\mathbb{Z})|=2$. A hint is provided: if $E=\mathbb{Z} \rtimes G$ then every element in $E - \mathbb{Z}$ has ...
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1answer
27 views

Finding inverse using one step subgroup test

$G=GL\left ( n,\mathbb{R} \right )$ $H= \{ A \in GL(n,\mathbb{R})|AA^{T}=I \}$ Is $H$ a subgroup of group $G$? $G$ is a group so $G$ contains the identity element. $A=I=e.$ Then, ...
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0answers
42 views

Homomorphisms from a $p$-group to $\mathbb{F}_p$

I'm doing a problem on group cohomology and have reduced it to the following: if $P$ is a $p$-group then $\textrm{Hom}(P,\mathbb{F}_p) \simeq P/\Phi(P)$ where $\Phi(P)$ is the Frattini subgroup of ...
3
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1answer
72 views

Homorphism from $B(G)$ to $\mathbb{Z}$

Let $G$ be a finite group, and $B(G)$ be its Burnside ring. Show that each ring homorphism $\varphi:B(G)\to\mathbb{Z}$ is the mark of some $H\le G$, i.e. it maps to an equivalent class of finite ...
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1answer
38 views

$G\oplus H$ is cyclic iff finite groups $G$ and $H$ are cyclic and $\gcd(|G|,|H|)=1$

Show that $G\oplus H$ is cyclic iff the finite groups $G$ and $H$ are cyclic and $\gcd(|G|,|H|)=1$ My answer is: $(\Rightarrow )$ Suppose that $G\oplus H$ is cyclic. Then there is a generator ...
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0answers
26 views

Nilpotent residual subgroup

Let $G$ be a finite group and $\gamma_{\infty}(G)$ be the limit of the lower central series: $\gamma_1(G)=G$ and for all $i\ge 1, \gamma_{i+1}(G)=[\gamma_i(G),G]$ ? It can be shown that ...
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1answer
21 views

A lower bound to the index of a subgroup of a non abelian simple group

Let G be a simple nonabelian group and $p$ is the largest prime number which divides $|G|$, prove that if $H$ is a subgroup of $G$ then $|G:H|\ge p$ I tried to show that if $|G:H|<p$ then $H$ is ...
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2answers
81 views

Proving that ${\rm Aut}(G)=\{{\rm id}\} \implies |G| \in \{1,2\}$. [duplicate]

The exercise asks to prove that if $G$ is any group with ${\rm Aut}(G) = \{{\rm id}\}$, then $g^2=1$ for all $g$ in $G$, $G$ is abelian, and if $G$ is finite, we'll have $|G| = 1$ or $2$. I managed ...
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1answer
37 views

What are $P\Gamma L(n,q)$ and $P\Sigma L(n,q)$?

I keep encountering the notations $P\Gamma L(n,q)$ and $P\Sigma L(n,q)$ in articles on group theory. I have more or less not been able to find references defining these groups, although this MSE ...
2
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1answer
39 views

Group of order $p^n$ has element of order $p$ without Cauchy's theorem

Let $p$ be a prime number and $G$ be a group with $|G|=p^n$. Show that G contains an element of order $p$. I would immediately say: "use Cauchy's theorem!", but this question is from a course that ...
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0answers
36 views

Isaacs FGT Problem 5B.1

I am trying to solve Isaacs' Theory of Finite Groups problem 5B.1. Let $G$ be finite and suppose that $P\in Syl_p(G)$ and that $g\in P$ has order $p$. If $g\in G'$, but $g\notin P'$, show that ...
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26 views

A $p$-group with certain lower central series

I am studying a paper in my research topic and I do not understand some facts in this paper. Let me explain the assumptions: Let $G$ be a finite $p$-group with the lower central series ...
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20 views

Pairs (m/n) , such that no non-trivial semidirect product of $\mathbb Z_m$ and $\mathbb Z_n$ exists

For which pairs $(m/n)$ is every homomorphism $\mathbb Z_m->Aut(\mathbb Z_n)$ AND every homomorphism $\mathbb Z_n->Aut(\mathbb Z_m)$ trivial ? Motivation : I want to find out, for which $m$ ...
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1answer
24 views

Proof for $\left \langle a^k \right \rangle=\left \langle a^{\gcd(n,k)} \right \rangle$

Theorem: $\left \langle a^k \right \rangle=\left \langle a^{\gcd(n,k)} \right \rangle$ Let $a$ be an element of order n in a group and let $k$ be a positive integer. Then $\left \langle a^k ...
2
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1answer
30 views

Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
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0answers
38 views

element of a group has finite order implies element is a generator?

Let G be a group and let a belong to G. If a has infinite order then all distinct powers of a are distinct group elements. If a has finite order, say, n, then $\left \langle a \right \rangle=\left ...
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1answer
21 views

generator for a cyclic group under addition modulo

Why is the generator for set $\mathbb{Z}_{n}=\left \{ 0,1,...,n-1 \right \}$ $1$ and $-1?$ The theorem says for a group $G$ to be a cyclic group there must exists an element a in the group G such ...
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1answer
36 views

Does Toeplitz matrices form a group?

In the wikipedia link of circulant matrix https://en.wikipedia.org/wiki/Circulant_matrix it is clearly written that "They can be interpreted analytically as the integral kernel of a convolution ...
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40 views

Is there an easy criterion for a finite group, for which every proper subgroup is abelian?

Is there an easy criterion for a finite group for the property that every proper subgroup is abelian (not necessarily a normal subgroup) ? It is clear that the abelian groups have this property, ...
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1answer
23 views

$p$- subgroups of $Gl_2(q)$

This question is similar to non-abelian groups of order $p^2q^2$., for which Derek Holt gave an answer to one of the cases, but I am looking for an answer that cover all the cases. I am looking for ...
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Name for the fact that a mattress can't be evenly rotated by repeatedly applying the same transformation?

Please excuse any errors in terminology or notation, I am neither a mathematician nor do I play one on TV. I'm pretty sure this is a known problem, probably named, but I lack the background knowledge ...
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37 views

Is there any generator with $\sqrt{|G|}\leq |X|\leq \frac{|G|}{2}$?

Let $G$ be a finite group of order $\geq 6$. Is there any generator $X$ for $G$ such that $\sqrt{|G|}\leq |X|\leq \frac{|G|}{2}$? If yes, what about $\sqrt{|G|}< |X|<\frac{|G|}{2}$ (if ...
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1answer
45 views

The relation between quasi-permutation matrix and permutation matrix?

We know that a quasi-permutation matrix is a square matrix over the complex numbers with non-negative integral trace. Can anyone tell me why it is called "quasi-permutation matrix"? Is there any ...
2
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1answer
59 views

On the element order of $GL(2,q)$

Is there a element of this type \begin{equation} \left( \begin{array}{cc} 0 & a \\ 1 & b \\ \end{array} \right) \end{equation} with order $2(q-1)$ in $GL(2,q)$, where $q$ is an odd prime? If ...
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0answers
39 views

If $|G_1|=|G_2|<\infty$ and $|G_1'|<|G_2'|$, then $|Z(G_1)|\geq |Z(G_2)|$? where $G'$ is the commutator subgroup of $G$.

We know that $G'$ characterization how ``abelian'' of a group because we have a theorem: if $G'=\{e\}$, then $G$ is abelian. I have a conjecture. If there are two finite groups $G_1$ and $G_2$, ...
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An equation in finite groups

Let $G$ be a finite group, $A$ a given subset and put and put $A^{-1}=\{ a^{-1}:a\in A\}$. We need a gap code for determining the maximum and minimum of all $|B|$ such that $B$ is a solution of the ...
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1answer
157 views

Does $A^{-1}A=G$ imply that $AA^{-1}=G$?

Let $G$ be a group, $A\subseteq G$ and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $A^{-1}A=G$ then $AA^{-1}=G$ (and visa versa)?
3
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1answer
27 views

Is this enough to justify an internal direct product?

I have a group $G$ and two subgroups $H,K$. Both $H$ and $K$ are normal in $G$, and $H\cap K = {id}$. Is this enough information to say that $G$ is the internal direct product of $H$ and $K$, or must ...
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2answers
33 views

Determining all elements of a certain order

Determine all elements of order 4 in $Z_{20} \times Z_{14}$ I'm not sure how to go about this other than just manually checking the elements? But I feel like there must be a better way? Thanks
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2answers
35 views

Pontryagin Dual of a Finite Abelian Group [closed]

Let $M$ be a finite abelian group. I want to show that the Pontryagin dual is a finite abelian group, and in particular I am interested in computing the elementary divisors/invariant factors of it. ...
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1answer
47 views

On finite capable $p$-group of class two

Do there exists a finite capable $p$-group $G$ of class two with cyclic center and the center is not subgroup of Frattini subgroup of $G$? A group $G$ is capable if there exists a group $H$ such ...
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1answer
66 views

Isomorphisms in GAP with large groups

So I decided to compute the first few terms of the automorphism series (finite part of the automorphism tower) for SmallGroup(16,3) in GAP in part to verify that $Aut^6(G)\simeq Aut^7(G)$ where ...
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1answer
46 views

Is there a change of basis that transforms any finite subgroup of $GL_n(C)$ into a subgroup of $GL_n(\bar{Q})$?

I vaguely know that there is a related statement that is true... something like, if G is finite, then every representation of it can defined over some finite algebraic extension F of Q. (By defined ...
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1answer
45 views

Isomorphic subgroups of finite groups

Which is the smallest number $n$ such that $S_n$ has non-isomorphic subgroups of the same order with the same number of cyclic subgroups of the same order? Example: $S_4$ has subgroups of ...
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1answer
68 views

What is known about this group reminiscent of the anharmonic group?

The anharmonic group is this nonabelian group of six rational functions with the operation of composition of functions: \begin{align} t & \mapsto t & & \text{order 1} \\[8pt] t & ...
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1answer
25 views

Generator of a cyclic group among generators.

Let $G$ be a group generated by the finite set $X=\{x_1,\ldots, x_n\}$. Now suppose that $G$ is a finite cyclic group. It is clear that $G$ need not be generated by $x_i$ for any $i$. What additional ...
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1answer
17 views

$k$-Normal subsets of finite groups

Let $G$ be a finite group of order $n\geq 6$ and $k$ a fixed integer with $1\leq k\leq n$. We define a $k$-property as follows: $gA=Ag$ for every $g\in G$ and every $A\subseteq G$ with $|A|=k$; (1) ...
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0answers
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Normal closure of a nilpotent subgroup

Is there a finite non-solvable group $G$ with a nilpotent subgroup $H$ satisfying its normal closure $H^G=G$? We guess that the answer is negative in the case that $H$ is S-semipermutable.($H$ is ...
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3answers
395 views

Does every non-trivial finite group have a subgroup with prime index?

Let G be any non-trivial finite group. Has G always a subgroup, whose index is prime ? If G is solvable and |G| has a prime divisor $p$, such that $p^2$ does not divide $|G|$, this is the case ...
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2answers
60 views

Abstract Algebra - Finite Group [closed]

Let G be a non-trivial finite group. For every $a,b \in G$ that are not identities, there exist $c \in G$ such that $b=c^{-1}ac$. Show that $|G|=2$.
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1answer
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The number of isomorphism classes for the symmetry group of 6 elements? [duplicate]

The different isomorphism classes of subgroups of $S_3$ is: trivial, $Z_2$, $Z_3$ and $S_3$ itself - that is $4$ different types The number of isomorphism classes of subgroups of $S_4$ is $9$ and ...
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2answers
34 views

G is a cyclic group of cardinality of a power of a prime number [closed]

Let $G$ be a finite group for which for every subgroups $H,K$ of it we have $H\subseteq K$ or $K\subseteq H$. Prove that $G$ is a cyclic group and its cardinal is a power of a prime number.
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0answers
19 views

Automorphism on $S_n$ sends given k-transposition to a transposition

For given integer $k>1$ and k-transposition $g \in S_n$, is there a isomorphism from symmetric group $S_n$ to itself such than sends $g$ to a transposition (a 2-cycle)? It seems that for $k=2$ it ...
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1answer
41 views

For which numbers $n$ is every group of order $n$ nilpotent?

I would like to have a simple criterion for the numbers $n$ with the property that every group of order $n$ is nilpotent. If $n$ is a prime power, it is clear that $n$ has this property. If $n$ is ...
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27 views

Which kind of product do we have here?

The following GAP-output ...
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1answer
22 views

Let A be a finite group and P be a normal p sylow subgroup. What is the connection between P and $Tor_p(A)$

Let A be a finite group and P be a normal p sylow subgroup. can there be an element $g \in A$ where $order(g) = p^x$ where x>0 and $g \notin P$ ? what I really try to understand is the connection ...
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1answer
44 views

Classification of subgroups of finite groups

For me the relevant number of subgroups of a finite group is the number of non-isomorphic subgroups. Mathematicians seems to have an other opinion. There is a related classification called ...
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1answer
42 views

Index of Subgroup in Alternating Group $A_n$

I am trying to show that for $n\geq 5$, the alternating group $A_n$ has no subgroup of index $p$ where $p$ prime and $p\not = n$. I am supposed to show this without using any of the Sylow theorems. I ...
3
votes
1answer
58 views

How can I prove that every finite product can be transformed to the given form?

Suppose, the permutations $a=(123)$ , $b=(12)(34)$ , $c=(12345)$ and $d=(12)(35)$ are given. I checked with GAP that the elemts $$a^jb^kc^ld^m$$ with $0\le j\le 2$ , $0\le k\le 1$ , $0\le l\le 4$ , ...