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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4
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1answer
86 views

Existence of non-abelian group of order n

I know this is an old quetion, but I've certainly been disappointed with the given answers. The question is: There exists a characterization of the natural numbers $n$ such that there exist at least ...
2
votes
1answer
40 views

The minimal normal subgroups of a maximal subgroup $L$ if two minimal normal subgroups of $G$ are not in $L$

If a finite group $G$ contains a maximal subgroup $L$ and two minimal normal subgroups not in $L$, then every minimal normal subgroup of $L$ is contained in the subgroup generated by the minimal ...
1
vote
1answer
50 views

Simple method to determine the sign of the permutation $x \rightarrow x^{-1}$ on a finite group

Let $G$ be a finite group. Let $f: G\rightarrow G$ be the map defined by $f(x) = x^{-1}$. Is there a simple method to determine the sign of the permutation $f$? The motivation is as follows(I ...
3
votes
1answer
41 views

What could be said about centralizers of normal subgroups if $G$ contains a simple, non-abelian maximal subgroup

Let $G$ be a finite group containing a maximal simple and non-abelian group, is it true that the centraliser of each normal subgroup is either trivial or a minimal normal subgroups? EDIT: Maybe to ...
1
vote
1answer
23 views

Congruence Class of Negative Integers in a Multplicative Group

As part of a larger problem, I need to find the subgroup of $(\mathbb{Z}/7\mathbb{Z})^*$ generated but the congruence class of -1. I understand that this is a multiplicative group with elements {1, ...
6
votes
1answer
193 views

Proving that a group of order $pqr$ (with conditions on those primes) is abelian.

I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ...
4
votes
1answer
35 views

All maximal subgroups are complement

Let $G$ be a finite group such that for any maximal subgroup $M$ and a subgroup of $H$, we have $MH=G$ or $MH=M$. Can we say something about this group ? Note that the equality is claerly satisfied ...
1
vote
1answer
34 views

If two quotient groups are semi-simple, then a third build from both is semi-simple too.

I call a group semi-simple if it is the direct product of non-abelian simple groups. Let $G$ be a finite group and let $M, N \unlhd G$ such that $G/N$ and $G/M$ are both semi-simple. Prove that ...
3
votes
1answer
53 views

For a subgroup $H$ of a finite group $G$ , when does $|Aut(H)|$ divides $|Aut(G)|$?

Let $H$ be a subgroup of a finite group $G$, then is it true that $|Aut(H)|$ divides $|Aut(G)|$? What if we also assume $G$ is abelian? (I know that $|Aut(H)| \space \big| \space |Aut(G)|$ if $G$ is ...
0
votes
2answers
48 views

Non-generator elements of a group and Intersection of all maximal subgroups.

A non-generator element $u$ of a group $G$ is defined as, If $H\not=G,$ then $\langle H,u\rangle\not=G$ for any $H\le G.$ Show that set of all non-generators of $G$ is a subgroup of the ...
1
vote
0answers
15 views

Finite subgroups of $O_4(\mathbb{Q})$

I have a problem with the classification of finite subgroups(up to isomorphism) of $O_4(\mathbb{Q})$ (or $GL_4(\mathbb{Z})$). I know about classification of $GL_2(\mathbb{Q})$. Maybe somebody knows ...
0
votes
1answer
122 views

Let $G$ be a group. Let $G/K$ be an abelian group. Prove that $C=\{e\}$.

I'm stuck at this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p ...
0
votes
3answers
34 views

direct product of cyclic and non-cyclic group together.

consider direct product of two finite groups, one is cyclic and the other one is not, is the direct product cyclic? if both groups are not cyclic,what we can say about direct product of them? I ...
0
votes
2answers
93 views

How are quaternions a finite set?

I'm having trouble understanding how Quaternions are a finite set when you can express a quaternion as Q = a + ib + jc+ kd, because a, b, c, d are $\in$ of $\Re$ would this not mean that the set is ...
0
votes
2answers
68 views

Group Theory: Showing that a subgroup is isomorphic to a product of groups

I have the following question, where the topic being tested is cosets, order and Lagrange's theorem: Suppose that every element $x$ in a group $G$ satisfies $x^2 = e$. Prove that $G $is abelian. ...
1
vote
1answer
23 views

Problem in permutation groups involving conjugates

I have to find a permutation $a$ satisfying $ a xa^{-1}=y$ where $ x=(12) (34)$ and $y=(56) (13)$ My attempt in solving the problem was- $$ a(12)(34)a^{-1}= a(12)(a^{-1}a)(34)a^{-1}= ...
1
vote
1answer
37 views

Automorphism group of a non_abelian p_group

Let G be a non abelian p_group. When is set of all automorphisms group of G a p_group?
2
votes
1answer
51 views

Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

I'm stuck on this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. What ...
2
votes
1answer
89 views

Is $A_{4}\times Z_2\simeq \langle g,h \mid g^{12},h^2,{gh}^{12}, gh=hg\rangle$?

Is $A_{4}\times Z_2\simeq \langle g,h \mid g^{12},h^2,(gh)^{12}, gh=hg\rangle$? In addition, is $\operatorname{Aut}(A_{4}\times Z_2)= \operatorname{Aut}(A_{4})\times ...
9
votes
1answer
139 views

How many different groups of order $15$ there are?

I wanted to share with you my resolution of this exercise. How many different groups of order $15$ there are? My resolution: We're looking for groups such that $|G|=15=3\cdot 5$. Then: $G$ ...
1
vote
1answer
38 views

Conditions for Nilpotency of inverse image of homomorphism.

Let $\varphi : G \to L$ be a homomorphism and $U \le L$. Under what conditions is $\varphi^{-1}(U)$ nilpotent, if $U$ is nilpotent? And a closely related question. If $UN/N$ is a nilpotent subgroup ...
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votes
0answers
17 views

Example such that $HN/N ~\mbox{char}~ G/N$ and $N~\mbox{char}~G$, but $H$ not characteristic in $G$

If $H/N$ is characteristic in $G/N$ and $N$ is characteristic in $G$, then $H$ is characteristic in $G$, a proof could be found here or here. The notation, i.e. speaking about subgroups $H/N$ implies ...
2
votes
1answer
81 views

How to compute the pointwise stabilizer subgroup of a fixed-point subspace?

Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$. Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ...
1
vote
1answer
51 views

Finite groups acting on strings.

Let $s = abcdandsoon.. \ \in \Sigma^*$. Let $|s| = n$ be the length of $s$. Consider all permutations of the positioned symbols that make up $s$, such that $s$ is fixed under the permutation. So if ...
2
votes
2answers
66 views

Question on Proof that $O_p(C/(C\cap F(G)) = 1$ for $C = C_G(F(G))$.

I have a question on the proof of a lemma about the Fitting subgroup, I mention all used facts: If $N \unlhd G$ and $A ~\mbox{char}~ G$ be a characteristic subgroup of $G$. Then i) $A$ is normal in ...
0
votes
2answers
38 views

cyclic group contain normal subgroup of prime index

Let $G$ be finite cyclic goup i wont to show that $G$ contain normal subgroup of prime index. A group G is cyclic if $G$=$ \langle a \rangle$, for some a$\in$$G$. A finite cyclic group of order n ...
1
vote
0answers
50 views

Question on proof that maximal normal abelian subgroup is self-centralising in nilpotent groups

The following is known about finite groups: (*) If $G/Z(G)$ is cyclic, then $G$ is abelian. Proposition: Let $G$ be a nilpotent finite group and $N$ a maximal abelian subgroup of $G$. Then $C_G(N) = ...
1
vote
2answers
44 views

What are the transitive groups of degree $4$?

How can I find all of the transitive groups of degree $4$ (i.e. the subgroups $H$ of $S_4$, such that for every $1 \leq i, j \leq 4$ there is $\sigma \in H$, such that $\sigma(i) = j$)? I know that ...
1
vote
1answer
52 views

How to prove that a certain set is a coset of a subgroup of a group of characters mod $m$?

I have the following question: Let $m>1$ be a positive integer and $G:=G(m):={(\mathbb{Z}/m\mathbb{Z})}^{*}$. Let $p\in\mathbb{P}$ with the property $p\nmid m$. Let $\widehat{G}:=\text{Char}(G)$ ...
2
votes
1answer
56 views

Generators of $SL(n,F),GL(n,F),SO(n,R),O(n,R),U(n,C)$

How can you describe Generators of $SL(n,F),GL(n,F),SO(n,R),O(n,R),U(n,C)$, where $F$ is a finite field, $R$ is real numbers and $C$ is complex numbers, $GL$ is a general linear group, $SL$ is a ...
2
votes
1answer
32 views

What could be said about $U,V$ if $UN = VN$ for some $N \unlhd G$.

Let $N \unlhd G$ and let $U, V$ be two subgroups, if $UN = VN$, is it possible that $U \ne V$ if i) $U$ and $V$ are not contained in $N$, and ii) if $U\cap N = V\cap N = 1$. Of course, if $U, V \le ...
1
vote
2answers
30 views

Orders of elements in alternating group $A_8$

I have an issue with a question from some homework for my introduction to group theory course. For which integers $d$ does the alternating group $A_8$ have elements of order $d$? So through some ...
-1
votes
1answer
29 views

properties of alternating subgroup?

I was wondering, is it true that if $Alt_n$ is an alternating subgroup of $Sym_n$ for $n>3$, $Alt_{n-i}\leq Alt_n$ for all $i<n$?
0
votes
0answers
103 views

Why is the number of conjugacy classes modulo 16 equal to the order for a finite group of odd order? [duplicate]

Let $G$ be a group of odd order $n$ and suppose $|Con(G)| = k$ ( Con(G) is the set of conjugacy classes of G), prove that $$k \equiv n \pmod{ 16}.$$ How do I proceed on this? Thanks.
0
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0answers
38 views

Properties of p-residue group

Related thread (definition of $O^{p'}(G)$) : does minimality condition imply normal p-sylow subgroup > Assume that $G$ is a finite group, and that $p$ is a prime number dividing the order of $G$. ...
0
votes
0answers
62 views

Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
0
votes
1answer
44 views

Automorphisms of B_n

Consider the Coxeter group of type $B_n$. This group, of order $2^n n!$, can be identified with the group of odd permutations of the set $\{\pm 1,\dots,\pm n\}$ and is thus isomorphic to the ...
0
votes
1answer
28 views

Proof about the Sylow $2$-subgroups of permutation group such that each element has at most two fixed points

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every element of $g \in G^{\#} := G \setminus \{ 1_G \}$ has at most two fixed points. Suppose further that ...
4
votes
1answer
54 views

About the construction of resolvents in Galois theory (over $\mathbb{Q}$ in $\mathbb{C}$)

I have to say that my question is quite long and I apologize for this. The main idea is that I would like to show how to construct resolvents for any transitive subgroup of the permutation group to ...
1
vote
1answer
26 views

Frattini subgroup and generating sets

Let $P$ be a finite $p$-group. It is easy to see that the cardinality of the smallest possible size of a generating set (say, $d(P)$) for $P$ equals the dimension of $P/\Phi(P)$ as vector space. Now ...
3
votes
1answer
40 views

Two Lemmata about permutation groups such that every element has at most two fixed points

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every element of $g \in G^{\#} := G \setminus \{ 1_G \}$ has at most two fixed points. Suppose further that ...
2
votes
0answers
42 views

does minimality condition imply normal p-sylow subgroup >

Assume that $G$ is a finite group, and $p$ is a prime number dividing the order of $G$. Let $\cal C=\cal C(H)$ be the following condition : "$H$ is a normal subgroup of $G$ and $|G/H|$ is coprime to ...
0
votes
1answer
37 views

which of the following options are true?

Let $G$ be a group, which are true? $G$ has a nontrivial centre $C$, then $G/C$ has trivial centre. If $G \not = 1$, there exists a nontrivial homomorphism $h: \Bbb Z\to G$. If $|G|=p^3$, for $p$ is ...
2
votes
1answer
39 views

Find the number of Sylow $p$-subgroups of $G$, if we know that $\lvert G\rvert=6$

Today I've been looking the third Sylow theorem. My professor did an example in class, so I tried to solve the example by myself and then compare what I did with the answer of my professor. The ...
1
vote
1answer
17 views

A sufficient criterion for a finite group to be a Frobenius group

Suppose that $G$ has a non-trivial proper subgroup $H$ such that the following holds: Whenever $1 \ne X \le H$, then $N_G(X) \le H$. Then $G$ is a Frobenius group with Frobenius complement $H$. ...
16
votes
2answers
485 views

The Weaver Android app $\rightarrow$ cute combinatorics problem

There's an Android puzzle app called "The Weaver". My question is why every level seems to be solvable in far fewer moves than one might naively think. Here's a link for people who want to play along ...
1
vote
1answer
68 views

Every group of order $5^8$ contains a normal subgroup of order $5^6$

I want to know what theorems/ideas are behind this proof and would appreciate explanation of a more general result too, (if one exists).
2
votes
1answer
101 views

About first Sylow Theorem proof

I've been struggling with the first Sylow theorem proof we were given in class. This is how my professor introduced us the theorem: First Sylow Theorem: Let $G$ be a finite group. Let $\rvert G ...
0
votes
1answer
30 views

Argument about the size of Frobenius kernel, question on derivation

The following discussion is from the textbook Finite Group Theory by Kurzweil and Stellmacher: Let $G$ be a permutation group on $\Omega$ and $|\Omega| > 1$. Then $G$ is a Frobenius group on ...
1
vote
0answers
37 views

Normalizer of a transitive subgroup in the symmetric group

Let $G$ be a finie group, and $H$ be a core-free subgroup of $G$ (that is to say, there is no nontrivial normal subgroup of $G$ contained in $H$). Denote by $\Omega$ the set of right cosets of $H$ in ...