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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2
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0answers
41 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le ...
1
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0answers
29 views

Characteristic subgroups of non-abelian $p$-group

It is a difficult problem to determine all the characteristic subgroups of a non-abelian $p$-group. But, then, I would seek for as many characteristic subgroups as we can, for small order $p$-groups. ...
0
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0answers
26 views

If $E := F^{\ast}(G) \cong Sz(q)$, then the elements from $G \setminus E$ act as field automorphisms of odd order on $E$

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every nontrivial element fixes at most two points. Let $E := F^{\ast}(G)$ be the generalised Fitting subgroup. ...
0
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1answer
30 views

Let $G$ be a group of order $p^2q^2$ , $p$ does not divide $q^2-1$ , $q$ does not divide $p^2-1$ , then is $G$ abelian? [closed]

Let $G$ be a group of order $p^2q^2$ ,where $p,q$ are primes , $p$ does not divide $q^2-1$ ,and $q$ does not divide $p^2-1$ , then is $G$ abelian ?
1
vote
1answer
35 views

Let a finite group $G$ have $n(>0)$ elements of order $p$(a prime) . If the Sylow p-subgroup of $G$ is normal, then does $p$ divide $n+1$?

Suppose $G$ is a finite group and $p$ is a prime that divides $|G|$. Let $n$ denote the number of elements of $G$ that have order $p$ . If the Sylow p-subgroup of $G$ is normal, then is it true that ...
1
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2answers
47 views

The meaning of 'integral powers' and how $\{2,3\}$ generates $\mathbb Z_6$

There is a line in my text This shows that all such products of integral powers of $a$ and $b$ form a subgroup of $G$... What does 'integral powers' mean here? Additionally, there is an ...
6
votes
3answers
115 views

Is a finite group action on a finite set determined by its fixed points?

Suppose I am given a finite group $G$, and a finite set $X$, and told that $G$ acts on $X$, but not told how. However, suppose for every subgroup $H\le G$, I am given the subset $X^H\subset X$ of ...
2
votes
0answers
33 views

A specific question on three statements in a paper about fixed point bounded groups, its interpretation and its usage w.r.t. the Sylow theorems

This is a rather specific post, but I hope nevertheless someone can help me. I am refering to a specific paper, namely K. Mayaard, R. Waldecker, Transitive permutation groups where nontrivial ...
2
votes
1answer
22 views

If centralizer of involution has twice odd order, then group has twice odd order

Let $G$ be a nonregular, transitive permutation group such that every nontrivial element fixes at most two points. Now suppose $U := G_{\alpha}\cap G_{\beta} \ne 1$ and suppose that $G_{\alpha}$ has ...
2
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0answers
39 views

How to choose set in Group Isomorphism Algorithm( Quasipolynomial time)

In the paper titled "On the $n^{log_2(n)}$ Isomorphism Technique" by Gary L. Miller, it is written A group is a binary operation * , satisfying 1) and 2) . 1) a)$ \exists! x(a*b = x)$ ...
0
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0answers
30 views

If the product of two elements from components of finite group is involution, then are the factors also involutions?

Let $G$ be a finite group and denote by $E(G)$ the subgroup generated by all components of $G$. Suppose $E$ is a component and choose $L \le E(G)$ such that $E(G) = E\cdot L$. Let $y \in E(G)$ be ...
2
votes
0answers
23 views

If stabilizer contains Sylow $2$-subgroup $S$ and another nontrivial subgroup $X$ fixing two points, then $X$ normalizes $S$

Let $G$ be a transitive, nonregular permutation group acting on $\Omega$. Suppose that $|\Omega|$ is odd, then $G_{\alpha}$ contains a full Sylow $2$-subgroup $S$ of $G$. Suppose that $G = ...
0
votes
1answer
14 views

Maximal subgroups of affine general linear groups

For an odd prime $p$, let $G$ be the affine general linear group of degree 1 over $\mathbb F_p$, i.e., the semidirect product of $\mathbb F_p$ with $\mathbb F_p^{\ast}$ where $\mathbb F_p^{\ast}$ acts ...
0
votes
1answer
36 views

Exponent of a group

Let $1\to A\to B\to C\to 1$ be a short exact sequence of groups. Are there any nice formulas relating the exponent of $A$, $B$ and $C$? Is it true that exponent of $B$ is the product of exponents of ...
0
votes
2answers
31 views

Calculate the commutator subgroup of $S_4$

So I have been tasked with calculating the commutator subgroup of $S_4$. As a warmup, I was able to calculate the commutator subgroup of $S_3$ through brute force calculations as there were only $6^2$ ...
0
votes
1answer
30 views

How to find all Subgroups of Generalised quaternion group

I am looking to find all subgroups of the following generalised quaternion group $$Q_{20}= \langle a,b: a^{10}=1,b^2=a^5,ba=a^{-1}b\rangle$$ the question asks for all the elements and their orders ...
3
votes
0answers
52 views

Frobenius determinant theorem

can anyone please recommend a paper or a book that gives a detailed proof of the Frobenius determinant theorem? I have read some few papers I saw online but their informations are not sufficient for ...
4
votes
1answer
37 views

Affine scheme obtained from (commutative) group algebra

Let $G$ be a finite abelian group (written multiplicatively), $R$ a commutative ring and let $R [G]$ denote the set of all formal linear combinations of elements of $G$ with coefficients in $R$. Then ...
1
vote
1answer
32 views

Does this “Index condition” always implies that subgroup is normal?

Let $G$ be a finite group and $p$ be the smallest prime dividing $o(G)$,let $q(>p)$ be the next prime that divides $o(G)$.Suppose $G$ does not have any subgroup of index $p$ and has a subgroup ...
0
votes
3answers
54 views

Special linear group of order 2 over field of order 3

Let G = SL(2, F$_3$) (group of matrices of determinant 1 over the field of order 3). Find |G|. Show that Z(G) is not {1$_G$}. Determine the number of Sylow 3-subgroups of G. What is the isomorphism ...
2
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0answers
52 views

Subgroups of order $12$ of $PSL(2,11)$

How can I prove that the group $PSL(2,11)$ has a subgroup of order $12$ isomorphic to $A_4$ and another subgroup of order $12$ isomorphic to the dihedral $D_{12}$? Thanks for the help!
4
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1answer
47 views

Example of a group $G$ which has two non-conjugate Hall subgroups

I want to find an example of a finite group $G$, which has two non-conjugate $\pi$-subgroups of Hall. From the Hall's theorem I know that I have to search in the class of non-solvable groups. So my ...
0
votes
2answers
30 views

When $|G|=105$ and has a normal Sylow $3-$subgroup, then $G$ is abelian.

$|G|=105=3\cdot 5\cdot 7$. We know that the Sylow $5-$subgroup $P_5$ must be normal and the Sylow $7-$subgroup $P_7$ must also be normal by simply counting elements. With the additional assumption of ...
6
votes
4answers
138 views

Prove that there is no element of order $8$ in $SL(2,3)$

Let $SL(2,3)=SL(2,\mathbb{F}_3)$. Prove that there is no element of order $8$ in $SL(2,3)$. My attempt: Let $A$ be a matrix in $SL(2,3)$. Then $A=U X U^{-1} $ for some invertible $U$ where $X$ ...
4
votes
1answer
47 views

Subgroup of $S_n$ with elements having cycles of any possible length [duplicate]

Let $G$ be a subgroup of the symmetric group $S_n$. Suppose that $G$ has the following property: given any positive integer $k\leq n$, there exists an element of $G$ having a cycle of length ...
0
votes
1answer
33 views

What is the number of Sylow 2-subgroups of the symmetric group $S_5$?

I am trying to find the number of Sylow 2-sbgroups of the symmetric group S5. As $ \lvert S_5 \rvert =120=2^3 \cdot 3 \cdot 5$. It has 2-SSG, 3-SSG, 5-SSG. But how to calculate there numbers? What is ...
1
vote
1answer
80 views

Number of Sylow 3-subsgroups of special linear group

I am aware that questions on this topic are around on this site, but they all seem to require information about the group that is not available to me in this problem. Consider the special linear ...
1
vote
1answer
70 views

How do I prove this seemingly obvious property of subgroups

The statement is the following: Given an abelian group $G=\langle a_1,...,a_t\rangle$, and a subgroup $H$ of $G$, we need at most $t$ elements to generate $H$; i.e. $H=\langle b_1,...,b_t\rangle$ for ...
5
votes
1answer
70 views

How can I check whether a given finite group is a semidirect product of proper subgroups?

Suppose, a finite group $G$ is given. I want to check whether there is a proper normal subgroup $N$ of $G$ and a subgroup $H$ of $G$, such that $G$ is the semidirect product of the groups $N$ and ...
0
votes
1answer
42 views

Selecting an element of order $q$ in $(\mathbb Z/p\mathbb Z)^*$?

Suppose $q\mid p-1$ where $p,q$ are two distinct primes. Also let $h\in[1,p-1]$, then compute $g=h^{(p-1)/q}$ mod $p$. If $g\neq1$, then does it mean that $g$ is of order $q$? If yes, then how? ...
3
votes
1answer
38 views

A question concerning an exercise from Tao Vu

This is the exercise 4.1.5 from Tao Vu Additive Combinatorics. $Z$ is a finite additive group with a fixed symmetric non-degenerate bilinear form $\cdot$ Define $e: \mathbb{R}/\mathbb{Z} \to ...
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0answers
37 views

Are there $5$ non-cyclic groups of order $2^n$ with an element of order $2^{n-1}\ $?

There are $5$ non-cyclic groups of order $128$ with an element of order $64$, namely $$C64 \times C2\ ,\ C64 : C2\ ,\ D128\ ,\ QD128\ ,\ Q128$$ Similar, there are $5$ non-cyclic groups of order ...
3
votes
2answers
52 views

How does a short exact sequence say something about a group?

I have a follow-up question to my question here: How are simple groups the building blocks? In that question I asked about what it means when we say that the simple (finite) groups are the building ...
1
vote
3answers
45 views

If $H<G$ ($G$ is a finite group) and $|G|=m|H|$, then for all $g\in G$ we have $g^{m!}\in H$.

If $H$ is a subgroup of a finite group $G$, and if $|G|=m|H|$, then for all $g\in G$ we have $g^{m!}\in H$. The question suggests I adapt the proof of Lagrange's Theorem in the book (Groups and ...
6
votes
1answer
57 views

Which finite simple groups contain $PSL(2,q)$ for some $q\geq 4$?

Which nonabelian finite simple groups contain $PSL(2,q)$ for some $q$? Obviously $PSL(2,q)$ themselves do. Also, as $PSL(2,4)\cong PSL(2,5)\cong A_5\subset A_n,\; n\geq 5$, alternating (nonabelian ...
1
vote
0answers
38 views

What do we know about the set of Primitive Dirichlet Characters modulu an integer $N$?

I was working on Dirichlet's theorem on arithmetic progressions and I have the following question: Is the set of all Primitive Dirichlet Characters modulus some given (and fixed ) integer $N$ a ...
0
votes
0answers
26 views

How can I prove that the elements form the symmetric group $S_4\ $?

How can I prove that the elements $$x^j\cdot y^l\cdot x\cdot y^k$$ with $x=(12)$ , $y=(1234)$ , $0\le j \le 1$ , $0\le l\le 2$ , $0\le k\le 3$ form the symmetric group $S_4$ ? Is should be enough ...
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0answers
70 views

When are groups subgroups of a same group?

Assume that $G_{1}$ and $G_{2}$ are groups and that $G_{1} \cap G_{2}$ has a group structure that makes it a common subgroup of $G_{1}$ and $G_{2}$. In other words, the set $G_{1} \cap G_{2}$ is a ...
0
votes
0answers
21 views

Wreath Products, Sylow subgroups

The problem is to determine the Sylow $p$ subgroup of $S_{p^n}$. I know that this is eventually the wreath product of n copies of $C_p$ (the p cyclic group), and I can do it for $S_{p^2}$, but could ...
2
votes
2answers
77 views

A non-abelian group with exactly four elements of order 5?

Find a non-abelian group with exactly four elements of order 5. I'm pretty new to group theory and the best I can think of is $D_4$ has 5 elements of order 2. What's a group that satisfies the ...
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0answers
42 views

So There is no simple group of order $2187$? [duplicate]

I have a group of order $2187=3^7$. I used Sylow's theorem and got Sylow 3-subgroup as my group. How do I continue?
1
vote
1answer
25 views

If $H$ fixes three points, then could the normalizer of $H$ induce an orbit of size two on the fixed points

Let $G$ be a transitive permutation group of degree $\ge 5$ acting such that every four-point stabilizer is trivial. Equivalently this means that every nontrivial element has at most three fixed ...
2
votes
1answer
41 views

The “ring of characters” of a finite group and its automorphisms

Let $G$ be a finite group and let $C(G)$ denote the set of characters of $G$ (in my representation theory course the values these characters take are in $\mathbb{C}$, but this is one point I'd like to ...
1
vote
1answer
23 views

If $A_6$ acts on set of size $45$, then every involution fixes five points

Let $G \cong A_6$ and suppose that $G$ acts as a transitive permutation group on a set $\Omega$ of size $45$. I want to prove that every involution fixes five points. Any ideas how this could be done? ...
3
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0answers
45 views

Prove that $\phi$ is class preserving automorphism

Let $G$ be a finite group and $\phi:G\to G$ be an automorphism of $G$ such that $\phi|_P=conj(g)|_P$ (restriction of some inner automorphism of $G$) where $P$ is any sylow $p$-subgroup of $G$, then is ...
3
votes
2answers
48 views

Presentation of $S_4$ with $3$ generators and easier relators

The presentation $$< x^4,y^3,z^2,yxzx^2,zxzy^2x^3,zyzy >$$ for $S_4$ is quite complicated. $S_4$ could be easily generated by $2$ elements, but I prefer $3$ generators with orders $2,3$ and $4$ ...
1
vote
1answer
51 views

Are there non-abelian groups with a “large” center?

Let $G$ be a non-trivial finite group with order $n$ and let $p$ be the smallest prime factor of $n$. Can the order of the center be $\frac{n}{p}$ ? I did not find an example with GAP until ...
1
vote
0answers
19 views

Let $PSL(3,q)$, $q$ odd. Then for $p \mid q(q^2 - 1)$ and $s \mid q-1$ ($p,s$ prime) we have a $p$-subgroup $X\ne 1$ such that $s$ divides $|N_G(X)|$

Let $G = PSL(3, q)$ with $q$ odd, the projective special linear group over a finite field of order $q$. Let $p, s$ be prime numbers. If $p$ divides $q(q^2 - 1)$ and $s$ divides $q-1$, then there ...
1
vote
0answers
25 views

If $PSL_2(2^k)$ acts on odd set such that $|\mbox{fix}(g)|\le 3$ for $g\ne 1$, then $G_{\alpha}$ has element of order $\frac{q-1}{\gcd(q-1,3)}$

Let $G \cong PSL_2(q)$ where $q = 2^k \ge 8$. Suppose $G$ acts as a transitive permutation group on $\Omega$ such that $|\mbox{fix}(g)| \le 3$ for nontrivial $g \in G$. Assume $|\Omega| = |G : ...
0
votes
0answers
31 views

Rank of Non-Abelian Quotient Group

Let $G$, $G'$ be two (non-Abelian) finitely generated free groups, and $H$, $H'$ be their finitely generated normal subgroups respectively. I want to know if the following statement is true: $G/H ...