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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1answer
30 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
29 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
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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
102 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
35 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
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0answers
53 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
40 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
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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
49 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
39 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
35 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
27 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
37 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
447 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
65 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
93 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
28 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
34 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
54 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
43 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 $\mathbb{1} \le V^{\otimes n}$?
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0answers
16 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
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1answer
42 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
40 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
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0answers
86 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
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1answer
32 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
34 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
44 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
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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
49 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
125 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
40 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
41 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 ...
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3answers
40 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?
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0answers
53 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 $p,p^2,p^3,p^4,p*q,..,q^4$. ...
7
votes
1answer
81 views

$S_{n+1}$ not isomorphic to subgroup of $S_n \times S_n$

I've been asked to prove that there is no injective homomorphism from $S_{n+1}$ to $S_n \times S_n$ for $n\ge4$. This seems to me to follow from the fact that $S_{n+1}$ cannot be recognized as a ...
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votes
0answers
40 views

Counter Example for Normality of Groups

Give a Counter Example to show that if $K \unlhd H$ and $H$ is a characteristic of $G$, then K need not be normal in $G$. I have no idea which type of groups I should look at? Any hint is ...
2
votes
1answer
33 views

Finite abelian group of type S must be $G=H\times H$

In the following paper by Delaunay http://delaunay.perso.math.cnrs.fr/heuristics_2.pdf, it is claimed that a finite abelian group comes with a unique non-degenerate alternating bilinear pairing ...
0
votes
1answer
19 views

The Glauberman -Thompson theorem

What is the Glauberman-Thompson theorem? I read in a paper, if $% N_{N}(Z(J(P))$ is nilpotent (where $P$ is a Sylow $p$-subgroup of $N$ and $N$ is a minimal normal subgroup of $G$), then by the ...
3
votes
1answer
46 views

Subgroups of free products of cyclic groups

Consider the free product $\mathbb{Z}_{3} \star \mathbb{Z}_{3}$. How would one determine the number of subgroups of this product up to isomorphism? It is routine for the case of the product ...
-1
votes
1answer
26 views

Finite group and Bézout's identity

Prove, using Bézout's identity, that if $n, a \in\mathbb{Z}$ are coprime then $\mathbb{Z}_n = \langle a\rangle$.
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votes
2answers
29 views

Homomorphism, showing that two mappings are the same.

Let $G$ be a group and suppose that every element of $G$ is a product of elements of some subset $X\subset G$ (i.e., $G$ is generated by $X$, so for each $g\in G$, $g=x_1^{\pm1}x_2^{\pm1}\cdots ...
0
votes
1answer
33 views

Number of non-isomorphic groups of order 21

Let $G$ be a group of order $21$. Find the number of non-isomorphic groups of order $21$ My solution: If the group $G$ is commutative,then $G$ can be expressed as a direct product of cyclic groups ...
0
votes
2answers
48 views

Size of |G:H| in set theory

Let $|G:H|$ be the set of left cosets for $x\in G$ and $xh\in xH$. It is stated that for all $g\in G$, $g$ is in exactly $1$ coset, as for $g,e \in G, ge=g\in gH$. But I don't then see how it isn't ...
1
vote
1answer
35 views

Dihedral group is supersolvable

I need to show that Dihedral group $D_n$ is supersolvable. My Approach : I think the existence of a normal chain $\{e\} = G_0 \leqslant G_1 \leqslant ... \leqslant G_n = G$ satisfying following ...
4
votes
2answers
55 views

Dimension of irreducible module divides the dimension of the algebra?

Fact: $\chi(1)$ divides order of $|G|$ where $\chi$ is an irreducible character of $G$. Above fact is equivalent to say that if $V$ is an irreducible $A=\mathbb C [G]$ module then $\dim(V)$ divides ...