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
39 views

Permutation groups and Sylow's theorem

Suppose we have primes $p$ and $q$ such that $q|(p-1)$, how do we show that $S_p$ contains a subgroup of order $pq$?
3
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1answer
30 views

The Sylow $2$-subgroups of $PSL(2, p^n)$ for $p \ne 2$. Question on derivation.

Consider $G = PSL(2, q)$ with $q = p^n$, i.e. the projective special linear group over a finite field of order $q$. Then $G$ could be considered as the group of all mappings $$ x \mapsto \frac{ax + ...
2
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1answer
23 views

General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
0
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0answers
66 views

Conjugacy classes in a group of order $p^aq^b$

Let $\Pi_e(G)$ denotes the set of all orders of elements (except the identity element) in a group $G$. Is there a group $G$ with the following properties: i) $|G|=p^aq^b$ ($p$ and $q$ are distinct ...
2
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1answer
31 views

If $A \lhd P$ and $A = C_P(A)$, then $|P:A|$ divides $(|A| - 1)!$

This is problem 1.D.10 in Isaacs, Finite Group Theory. Let $A$ be maximal among the abelian normal subgroups of a $p$-group $P$. Show that $A = C_P(A)$, and deduce that $|P:A|$ divides $(|A|-1)!$ ...
3
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2answers
70 views

Calculating the normalizer of a sylow p-subgroup

It seems that its explained pretty well online how to find a p-sylow subgroup, but Im having a hard time finding an explanation of how to find a p-sylow subgroups normalizer. Take a specific 2-sylow ...
3
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1answer
36 views

What non-abelian groups have a minimal permutation representation that acts transitively $\{1,2,\ldots, k\}$?

This question asks if a minimal permutation representation $\overline{G}$ of a group $G$ (that is, a subgroup $\overline{G} \le S_k$ is isomorphic to $G$ and $k$ is minimal with respect to this ...
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1answer
38 views

Show that $ \mathbb{Z}_6 \oplus \mathbb{Z}_6/ \langle (2,3) \rangle $ is or is not cyclic.

I am asked if $ \mathbb{Z}_6 \oplus \mathbb{Z}_6/ \langle (2,3) \rangle $ is cyclic or not. Work: Well the order of 2 in $\mathbb{Z}_6$ is 3 and the order of 3 in $\mathbb{Z}_6$ is 2. Thus, the ...
4
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1answer
143 views

Is $n=8{,}574{,}796{,}230$ the smallest squarefree number $n$ with $gnu(n)>10^6$?

The number of groups of order $n$ (gnu(n)) can be calculated by a closed formula, if $n$ is squarefree. I could calculate the values with GAP, additionally, I programmed a version in PARI/GP, working ...
1
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3answers
59 views

Is this induced action transitive?

Let $G$ be a group of order $n$ and let $k$ be a smallest integer such that we have a injection from $G$ to $S_k$. Denote $\bar G$ as a image of $G$. Is the action of $\bar G$ on $\{1,2,...,k\}$ ...
2
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1answer
52 views

Show that a group of order $33$ cannot only have elements of order $11$ and $1$.

Attempt: Assume $G$ only has elements of order 11 and that there exists $x \in G$ such that $o(x) = 11$. Then $x^{o(G)} = e$. $x^{33} = (x^{11})^3 = e \implies x^{11}$ has order $3$. Contradiction.
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0answers
33 views

If $H$ is subnormal in $G$ and its index is a $\pi$-number, then $O^{\pi}(G) \le H$?

Let $\pi$ be a set of primes, then we call $n \in \mathbb N$ a $\pi$-number if it only contains prime divisor from $\pi$. If $G$ is a finite group, a $\pi$-group is a group whose order is a ...
1
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2answers
83 views

Describe all groups of 8 elements

I try to find all the groups of 8 elements. I have found: $\mathbb{Z}_8$ $\mathbb{Z}_2 \times \mathbb{Z}_4$ $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ quaternions I don't understand ...
3
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0answers
65 views

What can we say about $D_{2n}$ and $D_n$ if $n$ is even?

We know that if $n$ is an odd natural number then the dihedral group $D_{2n}$ is isomorphic to the group $D_n\times \mathbb{Z}_2$ where $D_m$ is the dihedral group of order $2m$, which has ...
1
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2answers
58 views

Proving a group of order $77$ has a subgroup of order $7$ without Sylow theorem.

The question is Show if $G$ has order 77 then $G$ has a subgroup of order 7. Without using Sylow Theorems. Attempt sketch: Let $x \in G$. By Lagrange's theorem the order of $x$ is either $1, ...
5
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2answers
131 views

Rubik's Revenge Cube in GAP

I'm trying to create the Rubik's Revenge (4x4x4 cube) group in GAP . Take the following net of the 4x4x4 cube with each sticker labelled with a number. The front, left, upper, right, down, and back ...
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1answer
31 views

Frattini subgroup of a finite elementary abelian $p$-group is trivial

I would like to improve my proof of the following result: If $H$ is a finite, elementary abelian $p$-group, then $\Phi(H) = 1$. Here, $\Phi(H)$ is the Frattini subgroup, defined as the ...
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0answers
35 views

If $G / N$ is a $\pi$-group, then $O^{\pi}(N) = O^{\pi}(G)$?

Let $G$ be a finite group and $\pi$ a set of primes, then $O^{\pi}(G)$ denotes the smallest normal subgroup such that $G / O^{\pi}(G)$ is a $\pi$-group (i.e. a group such that its order is only ...
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1answer
50 views

Find the left cosets of subroups of $S_3$

So I am struggling to understand the definition of a coset. If I have the following symmetric group $S_3=\{1, \sigma, \sigma\tau, \sigma\tau^2, \tau, \tau^2\}$, where ...
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0answers
75 views

Is $gnu(n)<n$ always true for cubefree $n>1$?

Let $gnu(n)$ be the number of groups of order $n$. If $n$ is cubefree, (there is no prime $p$ with $p^3|n$), does the inequality $gnu(n)<n$ always hold for $n>1$ ? According to GAP, upto ...
1
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1answer
38 views

General formula to compute the exponent of the symmetric group $S_n$

Someone has already asked whether an exponent less than $n!$ is possible for a symmetric group $S_n$. It has been answered that it is for $n \ge 4$. I would like to know if there is a general ...
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1answer
41 views

Prove that a group of order 25 has a subgroup of order 5

I found this question Subgroup(s) of a group of order 25 I want to know if proving such a statement is possible by contradiction. Question: Let G be a group of order 25. Prove that G has at least ...
2
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1answer
85 views

$G$ is a finite abelian group. For every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$.

$G$ is a finite abelian group. Assume that for every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. I'd like to prove that $G$ is cyclic. I'm thinking about the approach of ...
1
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1answer
19 views

Prove the size of a finite group G divided by the size of the centralizer of a normal subgroup divides the size of that subgroup minus 1 factorial

Let $G$ be a finite group of size $n$. Let $H$ be a normal subgroup of size $m$. Let $C(H)$ be the centralizer of the $H$, with size $k$. Prove $n/k$ divides $(m-1)!$ I'm not really sure how to ...
0
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0answers
32 views

If $G$ acts so that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$. Conditions such $S \in \mbox{Syl}_p(G)$ has maximal class.

Let $G$ be a nonregular, transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Further suppose that for $g ...
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0answers
25 views

Importance of centralizer of involutions in finite group theory

I just read the chapter 1 of the book Finite Group Theory by John Rose, entitled "introduction to finite group theory". There, the author introduces the idea of the centralizer of an involution as an ...
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1answer
37 views

Given the subset A of S4 = {(1),(1 2),(3 4),(1 2)(3 4)}, how do I show that A is a subgroup of S4?

I know the subgroup test consists of being nonempty, being closed, and being closed under inverses, but I'm not quite sure how to apply it here. In addition, is S4 cyclic? Is it abelian?
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0answers
17 views

If $G$ is solvable and acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $M$ maximal, why is $N_M(M_{\alpha}) \in \mbox{Syl}_p(M)$?

Let $p$ be an odd prime. Suppose $G$ is solvable and acts as a nonregular and transitive permutation group on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points. ...
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0answers
36 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$, $M$ maximal with $|G : M| = p$, then $|M / L| = p$ for semiregular $L \unlhd G$.

Let $G$ be a solvable, nonregular and transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. And suppose that ...
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0answers
19 views

The uniqueness of the Frobenius representation as handled in textbooks, for example Kurzweil/Stellmacher

As written on Wikipedia:Frobenius_group The Frobenius kernel $K$ is uniquely determined by $G$ as it is the Fitting subgroup, and the Frobenius complement is uniquely determined up to conjugacy by ...
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1answer
33 views

Characterization of finite nilpotent group.

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. Is the following true : (1) G is nilpotent if and only if $N$ and $G/N$ are nilpotent. (2) Let $Z(G)$ be the center of $G$. Then $G$ is ...
0
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1answer
33 views

Let $g \in N_G(H)$ an element of order $5$. Compute the order of $H \langle g \rangle$.

Let $G = A_5$ and $H=\langle(1,2,3,4,5)\rangle$. Let $g \in N_G(H)$ an element of order $5$. Compute the order of $H \langle g \rangle$. I think I can use the second isomorphism theorem to solve ...
4
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4answers
97 views

Details about Caley's Group Theorem

The Caley-group-theorem states that every group is isomorphic to a subgroup of a permutation group. I am especially interested in the case that the group is finite. My question : If G is a group ...
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1answer
42 views

Show that character vanishes on specific element $g$ if for $H \le G$ we have $[\chi_H, 1_H] = 0$ and all elements of $Hg$ are conjugate in $G$

Let $G$ be a finite group and $H \le G$ with $g \in G$ such that all elements of the coset $Hg$ are conjugate in $G$. Let $\chi$ be a $\mathbb C$-character of $G$ such that $[\chi_H, 1_H] = 0$. Show ...
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0answers
52 views

Finding all the subgroups of a finite group [duplicate]

I need a method to find all subgroups from any finite group. For example, the subgroups of $D_4 $ (of order $4$): $\{(1), (1 \ 2 \ 3 \ 4), (1 \ 3)(2 \ 4), (1 \ 4 \ 3 \ 2)\}$. Okay, I can understand ...
0
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4answers
34 views

Cardinality of the smallest subgroup containing two distinct subgroups of order 2

$G$ is a finite group and $H_1$,$H_2$ are two disjoint subgroups of order $2$. $H$ is the subgroup of smallest order that contains both $H_1$ and $H_2.$ What is the ...
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0answers
62 views

Aut($\mathbb{Z}_2\times \mathbb{Z}_2)\cong \mathbb{Z}_6$

Would it be correct to prove this statement as follows? $(0,0)$ must always be sent to $(0,0)$ under any $\phi\in$ Aut($\mathbb{Z}_2\times \mathbb{Z}_2)$. Now, for Out($\mathbb{Z}_2\times ...
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0answers
57 views

Number of elements of order $5$ in group of order $5\times 13\times 43\times 73$

Let $G$ be a group of order $5\times 13\times 43\times 73$. Find the number of elements of order $5$. Here is what I do: Since $|G| = 5m$ where $(m,5) = 1$, $m = 13\times 43\times 73$, by Sylow's ...
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0answers
37 views

If $A$ is a semisimple algebra, and $M_1 \ncong M_2$ as irreducible $A$-modules, why we have that every ideal of $M_1(A)$ is an ideal of $A$

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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1answer
20 views

For a semisimple algebra and two $M$- and $W$-homogeneous parts for $M \ncong W$, why we have $M(A)W(A) = 0$.

Let $F$ be a field and $A$ be an $F$-vector space which is also a ring with $1$. Suppose for all $c \in F$ and $x,y \in A$ we have $$ (cx)y = c(xy) = x(cy) $$ Then $A$ is called an $F$-algebra. If ...
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1answer
91 views

Every group with 5 elements is an abelian group

I have tried to prove that every group with 5 elements is an abelian group using following approach. Is this correct: Note: I do do not want to use Lagranges theorem and I do not know why groups with ...
5
votes
3answers
189 views

Why group of order 6 has to have just two elements of order 3

In an attempt to prove that every group $G$ of order 6 is isomorphic to either $\mathbb{Z}_6$ or $S_3$, I stumbled upon one peculiar issue. We can use Cauchy's Theorem to argue that since ...
2
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1answer
33 views

What can we infer if the commutator subgroup $G'$ of a group $G$ is equal to $G$?

I know that if the commutator subgroup G' of a group G is equal to {1} then G is abelian. I think that if G' = G then for all a,b in G with a and b not equal to the identity ab is not equal to ba. ...
6
votes
1answer
61 views

How many groups of order $n$ with center {e} exist?

For which numbers $n$ exists a group of order $n$ with center {e} ? And how many groups are there for a given order ? The first such numbers are $6,10,12,14,18,20,21,...$ Groups of order ...
5
votes
2answers
93 views

Proof containing abelian groups.

Let $G \leq S_{999}$ be an abelian subgroup of order $|G| = 1111$. Prove that there exists $i \in$ {$1,2,...,999$} such that $\forall α \in G, α(i) = i$. Okay so I came across this problem and even ...
4
votes
2answers
134 views

Small version of the GAP-software or online tool available?

If I only want to calculate the number of groups of order $n$ for large $n$ , especially cubefree $n$ : Is there a small version available or an online calculator ? Or do I simply have to download ...
1
vote
1answer
40 views

Create a base that contains $x$ in a finite generated Abelian group

Let $A$ be a finite generated free Abelian group and $x \in A$ such that $\forall y \in A$ $\forall n>1: x\neq ny $. Prove that there is a base generating $A$ that contains $x$. It seems simple ...
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0answers
26 views

Size of the intersection of a $p$-Sylow subgroup and a normal subgroup.

Assume $G$ is a group whose size is $(p^r)*m$, where $p$ is prime and $p$ doesn't divide $m$. Let $P$ be a $P$-Sylow subgroup of $G$, and $H$ a normal subgroup of $G$. Lagrange's theorem gives us that ...
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0answers
27 views

Affine geometry and its basis

I am trying to solve an exercise from the book "Permutation Groups" by J. Dixon and B. Mortimer, but, this is not a homework. The affine geometry $AG_d(F)$ consists of points and affine subspaces ...
0
votes
2answers
37 views

If $G$ acts such that $\mbox{fix}(g) \in \{0,p\}$ for $g \ne 1$ and $N \unlhd G$. Then every element outside of $N$ fixes at most $p$ $N$-orbits.

Let $G$ be a transitive permutation group acting on $\Omega$ such that each nontrivial element either fixes no point or exactly $p$ points for some prime $p$. Also assume that for $g \notin ...