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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1answer
92 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
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1answer
45 views

Is my proof correct? (about commutators)

I want to prove the following fact: Let $G$ be a finite group and let $x, y\in G$ such that $[x, y] \in Z(G)$. Then $[x, y^s] = [x, y]^s$ for every $s\in \mathbb{Z}$. If we assume that $[x^r, y^s] \...
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1answer
77 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
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2answers
65 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...
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1answer
113 views

Show that the symmetric group $S_p = <\sigma , \tau >$, where $\sigma$ is any transposition and $\tau$ is any p- cycle and p is a prime number.

Let $\sigma = (a_1\ a_2) \ \ and \ \ \tau = (a_1\ b_2\ \ldots\ b_p)$. (We have $a_2 = b_i$ for some i.).We know that $S_p$ is generated by $\{ (a_1\ a_2) \ \ and \ \ (a_1\ a_2\ \ldots\ a_p) \}$. So ...
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4answers
110 views

multiplication in Galois Fields

I don't know much about Galois fields. My question is the following Assume we are working with GF(8). Let say for example I want to multiply 2 by 4 in GF(8). Then it should be equal to $2*4 \text{ ...
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1answer
78 views

Group elements $x$ and $y$ satisfying $x^2 = y^2x^2y$ and $yx^{-1}y^2 = x^7$ commute.

The Question Suppose that $x$ and $y$ are elements of a group such that $$x^2 = y^2x^2y$$ and $$yx^{-1}y^2 = x^7.$$ Show that $x$ and $y$ commute. Motivation This came up in another question, where ...
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2answers
141 views

Prove that there is a basis of a lattice $\Lambda$ s.t. a reflection is of a certain form

Consider $\Lambda$ a lattice in $\mathbb{R}^2$. Let $S \in O(\Lambda)$ be a reflection, i.e. $\det S = -1$. Set $S_{1}= \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$ and $S_{2}= \begin{...
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1answer
58 views

Lower central series and the center of $S/S^k$

Let $S$ be a $p$-group and assume that $S$ has maximal class - so if we assume $\vert S \vert = p^n$ then the lower (and upper) central series has length $n-1$. I don't know much about lower central ...
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1answer
75 views

what are the other 2 nontrivial elements of the automorphism group of $\Bbb Z/5\Bbb Z$?

It is known that the automorphism group of the units of $\Bbb Z/5\Bbb Z$ is isomorphic to the cyclic group of order $4$, so the automorphism group must also have $4$ elements. The two nontrivial ones ...
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0answers
31 views

Is there a special name for this single element discrete subgroup of Mobius group

Is there a special name for this discrete subgroup of Mobius group with single element: $$A=\begin{bmatrix}0 & i\\-i & 0\end{bmatrix}$$ and $\det A=-1$ and $A^2=I$. Thanks- mike EDIT: $$A=...
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2answers
296 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
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1answer
48 views

Finite groups with a cyclic maximal subgroup.

In the book A Course in the Theory of Groups by Derek J.S. Robinson, Finite $p$-groups with a cyclic maximal subgroup are classified. Now I wish to know whether finite groups with a cyclic maximal ...
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1answer
69 views

Inconsistent definition of Sylow p-subgroup

Here is the definition of a Sylow $p$-subgroup from Wikipedia: For a prime number $p$, a Sylow $p$-subgroup (sometimes $p$-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a ...
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0answers
111 views

Cokernel of injective endomorphisms of a finitely generated free abelian group

By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let $F_A=\mathbb{Z}^n/...
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0answers
81 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
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0answers
52 views

Prove that both $R_p$ and $R^p$ are abelian groups under ordinary addition of rationals.

Q: Let $p$ be a fixed prime. Let $R_p$ be the set of all those rational numbers whose denominator is relatively prime to $p$. Let $R^p$ be the set of rationals whose denominator is a power of $p\ (p^i,...
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1answer
113 views

Commuting Elements in a Free Product of Cyclic Groups

In the free group with two generators $F_2\cong\mathbb Z *\mathbb Z$ ($*$ denotes the free product), if two elements $a$ and $b$ commute, then there exists an element $w\in F_2$ such that $\langle a,b\...
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2answers
161 views

Show that $\mathbb{Z}_p\setminus\{\overline{0}\}$ is not a group if $p$ is not prime.

The answer is too short that I think I've gone wrong at some point! Q: If $p$ is prime, then the nonzero elements of $\mathbb{Z}_p$ form a group of order $p-1$ under multiplication. Show that this ...
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1answer
81 views

Density of nilpotent numbers

A natural number $n$ is nilpotent if every group of order $n$ is nilpotent (equivalently, a direct product of Sylow subgroups). A natural number $n$ has nilpotent factorization if $\ell\not\equiv1$ ...
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1answer
106 views

(Theorem) If $G$ is a simple group of odd order , then $G \cong \mathbb Z_p$ for some prime $p$.

I am studying Dumit Foote. I have seen this result in this book. Please help me solve this. Thank you.
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1answer
44 views

A question on the intuition of decomposition of the element of symmetry group

Any element of symmetry group $S_{n}$ can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there ...
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2answers
108 views

Number of conjugacy classes in finite groups

Let $G$ be a finite group. Let $C_1,C_2,\dots,C_k$ be its conjugacy classes. We denote by $C_{j\ '}=\{g^{-1}|\ g\in C_j\}$ the conjugacy class inverse to $C_j$. Set $$a_{rst} = \frac{1}{|C_t|}|\{(x,...
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1answer
51 views

I need an example of a function with these properties.

I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. these functions ...
0
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1answer
66 views

Action of factor group on a group

Suppose $A$ and $G$ are finite groups and $A$ acts on $G$, written $g^a$ for $g\in G$, $a\in A$. If $N\unlhd A$, does $A/N$ then act on $G$ also? By $g^{[a]} = g^a$? Or do I need to assume something ...
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1answer
49 views

Group theory problem

I am asked to prove "Show that if $${e}<H_1<H_2<...<H_{n-1}<G$$ Is a subnormal series for a group G, and if the order of $H_{i+1}/H_i=s_{i+1}$, then G is of order $s_1 s_2...s_{n-1}$...
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1answer
43 views

Group of order 105 as a product

I can't solve this problem, please help me! Every group of order 105 is isomorphic to $\mathbb{Z}_5\times H$ where $H$ is a group of order 21.
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1answer
95 views

For a group $G$ acting transitively, is $G_{\alpha}$ contained in stabilizer of some block $\Delta$ if $\alpha \in \Delta$?

This question refers to permutation groups, in particular the primitive ones, and block systems. Let $G$ be a finite group acting on a set $\Omega$, and consider some partition $\Delta_1 \cup \ldots \...
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3answers
639 views

Quaternion Group as Permutation Group

I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it. So I googled to see if somebody else had put the ...
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2answers
138 views

What is the mean prime power?

I found this definition in a book and I did not understand the meaning of it There exists a field of order q if and only if q is a prime power (i.e., $q = p^r$]) with p prime and r ∈ N. Moreover, if q ...
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0answers
35 views

Version of Chevalley's Theorem proving $\mathbb{F}_q$ is a $C_1$-field for $\mathbb{Z}/p^n\mathbb{Z}$?

Let $k=\mathbb{F}_q$ Chevalley's theorem states that if $f(x_i)\in k[x_1,...,x_n]$ is such that $f(0,...,0)=0$ and $deg(f)<n$, then there is a non-trivial $a_i\in k^n$ such that $f(a_i)=0$. Is ...
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1answer
92 views

Is there a finite non-solvable group which is $p$-solvable for every odd $p\in\pi(G)$?

Let $G$ be a finite non-solvable group and let $\pi(G)$ be the set of prime divisors of order of $G$. Can we say that there is $r \in \pi(G)-\{2\}$ such that $G$ is not a $r$-solvable group?
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1answer
115 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
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1answer
44 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
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0answers
24 views

structure of a p-group

Let $G=\langle x_1,\ldots,x_6\rangle$ be a non abelian group of order $p^6$ and exponent $p$. Also we know that $[x_i,x_j]=1$, for $1\leq i, j\leq6$, except $[x_1,x_5]=x_3$, $[x_1,x_6]=x_4$ and $[x_2,...
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1answer
106 views

Subgroup contained in all other subgroups

This is Problem 2.13.10 from Herstein, Topics in Algebra: Let $G$ be a finite abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H\neq (e)$. Prove that $G$ ...
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1answer
65 views

Elements of orders $2k$, for $k\geq 5$ in a semidirect product

Let $G$ be a non-solvable group, $N$ be an abelian 2-subgroup of $G$ such that $N=C_G(N)$ and $G/N\cong Sz(8)$. Does $G$ has elements of orders $2k$, for $k\geq 5$?
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1answer
46 views

Normal subgroups of factor group

Suppose $G$ is a finite group and $N$ is a normal subgroup of $G$. Then subgroups of $G/N$ are of the form $A/G$ for $A\le G$. But how does the normal subgroups of $G/N$ look like? Is it true that $A \...
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1answer
76 views

torsion and torsion-free groups

I have the following statement: Every finitely-generated abelian group $G$ is isomorphic to $T\bigoplus F$, where T and F are torsion and free groups. As an example is given that all abelian groups ...
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1answer
179 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq \{e\}...
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2answers
28 views

Question on Subgroups and Left Cosets

Here is an unanswered exercise from class notes. For the group <{$i,-i,1,-1$},*> and a subgroup H where $|H|$ = 2. Find the left cosets of G induced by H. G={$i,-i,1,-1$}
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3answers
82 views

Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
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118 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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1answer
98 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
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79 views

representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
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0answers
47 views

How can we find element orders of a finite group?

Suppose $p$ and $q$ are two prime divisor of the order of a finite group $G$. I want to know if $G$ has an element of order $pq$ using the character table of $G$. Is this possible? If so, please ...
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1answer
119 views

Do these definitions make sense?

Letting $G$ be a group and $S(G)$ be all permutations of $G$, define $$L(G)=\{\phi\in S(G)|\forall n\in \mathbb Z , \phi(g^n)=\phi(g)^n\ \ \forall g\in G \}.$$ It is easy to check that $L(G)$ is a ...
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1answer
65 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
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3answers
97 views

How is this subgroup abelian?

Let $G$ be a finite group of order $2n$ such that half of the elements of $G$ are of order $2$ and the other half form a subgroup $H$ of order $n$. Then I know that $H$ is of odd order because for ...
2
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0answers
56 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...