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
59 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
47 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 ...
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1answer
38 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.
2
votes
1answer
34 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
103 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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0answers
18 views

Frobenius subgroups of $GL(n,2)$

I need to know the structure of Frobenius subgroups of $GL(n,2)$, where $n<14$. Is there any good reference? Many thanks.
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2answers
43 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
20 views

What is the flag variety of the finite unitary group?

$G(q)$ finite Chevalley group is quasisplit, so there is a Borel subgroup $B(q) \subset G(q)$. $GL_n(q)/B(q)$ is the set of complete flags in $GF(q)^n$. What is $U_n(q)/B(q)$, where $U_n(q) \subset ...
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0answers
30 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 ...
4
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1answer
83 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
84 views

Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
3
votes
1answer
101 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 $p$. ...
2
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1answer
30 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 ...
2
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0answers
20 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 ...
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0answers
61 views

Direct product of $G'$ and $Z(G)$ with some conditions

Let $G=G'\times N$ be a non-solvable group, such that $G/N$ is a non-abelian simple group and $N=Z(G)<C_G(N)$ is an abelian normal minimal $p$-subgroup of $G$. What can we say about $G$? In a ...
2
votes
1answer
81 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
61 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 $k\geq 5$?
2
votes
1answer
30 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
24 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 ...
7
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1answer
136 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 ...
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2answers
23 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
51 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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0answers
70 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 $p$, $G$ ...
1
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1answer
56 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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0answers
30 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 ...
3
votes
0answers
39 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 ...
4
votes
1answer
117 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
51 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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2answers
53 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
votes
0answers
31 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 ...
3
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1answer
93 views

groups of order $p^5$ and exponent p

We know that a group of order $p^5$ is an extra special group. How we can show it doesn't have any abelian subgroup of order $p^4$? Also what is the presentation of this group if its exponent is equal ...
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1answer
56 views

Generating a subgroup

While getting ready for my exams I am trying to solve this question "Find a Sylow $2$-subgroup in $S_4$ which contains the element $(1, 3, 2, 4)$." I started by permutation of the element required ...
3
votes
3answers
51 views

Is a finite group with a certain automorphism must be abelian

Let $G$ be a finite group, and let $f:G \rightarrow G$ be an automorphism, such that $x f(x) f(f(x))=1$ for any $x \in G$. Is $G$ must be abelian? I believe that there are examples where $G$ is not ...
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2answers
24 views

A group of order $56$ with a unique Sylow $2$-group is either nilpotent or its Sylow $2$-group is $\cong (\mathbb{Z}/2 \mathbb{Z})^3$

Let $G$ be a group of order $56$, $Q$ a normal Sylow $2$-group and $P$ a Sylow $7$-group. Show either $G \cong Q \times P$ or $Q \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times ...
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0answers
32 views

For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
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2answers
160 views

Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
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1answer
49 views

Groups of order 8 proof

I understand the solution to these questions I was just wanting to confirm that the solution to Q10 excludes the possibility that $y =x^2$ ( and hence the proof is not complete) as it uses the ...
1
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1answer
49 views

Commutators Calculus

I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas? Definitions By recurrence we define $[x,_0\, y]=x$; ...
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1answer
71 views

Prove that $G$ has a nontrivial normal subgroup

I wanted to ask if I had done this problem correctly. Let $G$ be a group of order $pqr$ (for $p > q > r$ primes). (i) If $G$ fails to have a normal subgroup of order $p$, determine the ...
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0answers
19 views

Constructing all inequivalent faithful irreducible projective representations of finite abelian groups

Let $G$ be a finite abelian group, and $\alpha \in H^2 (G,\mathbb{C}^*)$ a 2-cohomology class. It is known (in Karpilovsky's multi-volume tome or elsewhere) that a finite abelian group admits a ...
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0answers
47 views

Normally embedded subgroups reducing in a Hall system

A Hall system of $G$ is a set $\Sigma$ of Hall subgroups of $G$ satisfying the following two properties: -For each $\pi$ divisor if $|G|$, $\Sigma$ contains excatly one Hall $\pi$-subgroup $G_{\pi}$. ...
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2answers
141 views

Is a finite group always a element-wise product of Sylow subgroups?

Let $G $ be a finite group, and let $ p_1, \ldots, p_n $ be the distinct primes dividing $|G|$. For each $i $, let $ P_i $ be a Sylow $ p_i $-subgroup of $ G $. I seem to recall a theorem saying $ ...
2
votes
1answer
20 views

Hopefully easy Lang-Steinberg computation: for Weyl elements

How do you write an element of the Weyl group as $g^{-1} F(g)$? For instance, let $G = \langle x_1(t), x_2(t), x_{-1}(t), x_{-2}(t) : t \in K \rangle$ where $K$ is an algebraically closed field ...
4
votes
1answer
68 views

Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
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2answers
40 views

Operations in finite field $F_p$

If I have a finite field $F_p$, where $p$ is prime how can I define operations like $+, -, \times, / $? Can I just make: $$add: (a + b) \mod p$$ $$sub: (a-b) \mod p$$ $$mul: (a\times b) \mod p$$ ...
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1answer
55 views

Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?

Question as stated in the title: Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$? If not, can you give me a counterexample? Thanks
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1answer
45 views

Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
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0answers
40 views

Exact sequences of $1 \to A \to SO(N) \to B \to 1$, special orthogonal group

Inspired by the nice post and this, apart from SU(N), now I am particularly looking into the exact sequences of SO(N), but I will like to loosen the conditions of the previous post, Q1. $$1 \to A ...
7
votes
1answer
123 views

Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian

Suppose that $G$ is a finite group. If $P( ab=ba ) >5/8$, prove $G$ is abelian.
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2answers
45 views

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic.

Let $G=\mathbb Z_4 \times \mathbb Z_2$. Find all $H$ subgroups of $G$ of order 2, so that $G / H$ is cyclic. I tried going brute force, and realized there are 3 candidates: $H_1=\{(0,0), (2,0) \}$ ...