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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9
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1answer
133 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
37 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 ...
0
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
77 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
64 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
34 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
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0answers
40 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
36 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
50 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
44 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
votes
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
59 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
42 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
27 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
52 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
40 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
29 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
38 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
468 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
98 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
29 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
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0answers
36 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 ...
0
votes
1answer
56 views

Example of three-generator abélien by cyclic

Can you give me example of three-generator group abelien by cyclic(i.e there exist normal subgroup $N$ in $G$ abelien and $G/N$ is cyclic) which is not finite by nilpotent (i.e there isn't finite ...
1
vote
2answers
44 views

The set of non-conjugate elements

I have $H \leq G$ where $G$ is a group. Now for any $t \notin H$ we have $H \cap tHt^{-1} = e$ Now $N$ is a set of all elements of $G$ which are not conjugate to any element of $H$ I want to ...
2
votes
2answers
37 views

Is the trivial representation a subrepresentation of a tensor power of any irreducible complex representation of a finite group?

Let $G$ be a finite group, $V$ an irreducible complex representation and $\mathbb{1}$ the trivial representation. Question: $\exists n >0$ such that $\mathbb{1} \le V^{\otimes n}$?
0
votes
0answers
19 views

Normal minimal supersolvable group

Let $G$ be a finite group, and let $N$ be minimal normal supersolvable subgroup of $G$. Why $N$ is elementary abelian group?
6
votes
1answer
44 views

Let $G$ a finite group such that $\lvert G \rvert=pm$, with $p$ a prime and $\gcd(p,m)=1$. $G$ has an unique Sylow $p$-subgroup $P$. Prove $P\lhd G$.

I just made this exercise, left as homework, and I'm almost sure that I did something wrong, or at least that there's a better way to solve it. Here it goes: Let $G$ a finite group such that ...
2
votes
1answer
22 views

Number of non-isomorphic groups of order $p^n$ where $p$ prime is equal to the number of partitions of $n$

Number of non-isomorphic groups of order $p^n$ where $p$ prime is equal to the number of partitions of $n$ :By a partition of $n$ we shall mean $n=n_1+n_2+...+n_k;n_1\geq n_2\geq ..\geq n_k>0$ My ...
3
votes
1answer
41 views

Prove that $\langle S\rangle$ $= G$, with $S \subset G$ and #$S > 1/p $ #$G$

I'm having trouble with solving this problem: Let $G$ be a finite group of order $> 1$, and $S \subset G$ a subset of $G$, with #$S > 1/p $ #$G$ Where p is the smallest prime factor of the ...
2
votes
0answers
94 views

Basic Survival Facts in Finite Group Theory [closed]

What are some basic facts of finite group theory used over and over again, so that pro's does not even think about them anymore (and even implicitly contained in many arguments). I mean facts ...
1
vote
1answer
37 views

On the permutability of two subgroups of coprime indexes

Let $G$ be a finite group and let $H$ and $K$ two subgroups of coprime indexes. Is it true that $G=HK$. Obviously $G=\langle H,\, K\rangle$ but why the permute each other? Any ideas?
0
votes
0answers
17 views

A finite non-$2$-nilpotent $\{2,3\}$-group

Let $G$ be a finite non-$2$-nilpotent $\{2,3\}$-group such that $G=PQ$, where $P$ is a normal elementary abelian Sylow $2$-subgroup of order $4$ in $% G$, and $Q$ is a cyclic Sylow $3$-subgroup of ...
2
votes
1answer
38 views

Finite Group is the Galois Group of an extension K/F

How can I show that every finite group is the Galois group of an extension K/F where F is itself a finite extension of Q? I know the following: (1)Every finite group is contained in $S_p$ for a ...
6
votes
1answer
45 views

Confusion with Lang's proof of Sylow Theorem

I am currently working through Lang's Algebra. I am rather confused by what seems to be a trivial point. In a lemma preceding the proof of the Sylow Theorem (which is essentially Cauchy's Theorem), ...
1
vote
1answer
62 views

Every finite group is contained in $S_p$

I am reading about the inverse Galois problem. I stumbled with the problem of showing that every finite group is contained in $S_p$ for a large enough prime $p$, is this true? does anybody have a hint ...
0
votes
1answer
50 views

Prove: $<S>$ $= G$, and every $x \in G$ can be written as $x = s_{1}s_{2}$ with $s_{1}, s_{2} \in S$

I'm trying to solve this problem for my math study, but the things I'm trying don't seem to work. Let $G$ be a finite group, and $S \subset G$ a subset of $G$, with #$S > 1/2 $#$G$ Prove: a) ...
3
votes
1answer
127 views

Computing character of a representation and irreduciblity

for a finite field $k$ I have $G = SL_2(k)$ a group. $H \leq G $ and $H = \lbrace $ $\begin{bmatrix} a & b \\ 0 & d\\ \end{bmatrix} \vert a,b,d \in k \rbrace $ Now $\omega : k^{*} ...
3
votes
1answer
41 views

To show that $A_4$ is solvable

I need to show that $A_4$ is solvable. From what i know the definition of solvable expects to give some chain of subgroups such that each subgroup in the chain is normal to the one in which it is ...
3
votes
0answers
51 views

Maximal abelian subgroup of general linear groups

Thanks for any help or comments. Is it possible to recognize all maximal abelian subgroups of general linear group on finite field $F$ of order $q$, $GL_n(F)$. By maximal abelian I mean if $A$ is ...
1
vote
3answers
44 views

Does $P \in Syl_p(N_G(P))$ implies $P \in Syl_p(G)$?

In a finite group $G$, let $P \in Syl_p(N_G(P))$, i.e. $P$ is a Sylow $p$-subgroup in its normaliser, does this imply $P \in Syl_p(G)$, i.e. it is a Sylow $p$-subgroup in its entire group?
1
vote
0answers
59 views

How many nonabelian groups up to isomorphism are of the order $p^4q^4$?

For distinct primes $p$ and $q$, how many nonabelian groups up to isomorphism are of the order $p^4*q^4$? We can say that there are nontrivial subgroups with cardinality $p,p^2,p^3,p^4,p*q,..,q^4$. ...