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Questions tagged [finite-groups]

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.

0
votes
1answer
30 views

Showing a subset is a subgroup

Let $A$ be a subset of a finite group $G$. For any $g\in G$ we denote by $gA$ the set $\{ga : a ∈ A\}$. Assume that $e \in A$, where $e$ is the identity element of $G$, and that for any $g_1, g_2 \in ...
6
votes
1answer
65 views

How do basic results in representation theory change going to positive characteristic fields and/or not algebraically closed?

I'm working on a project linking graph theory, model theory and representation theory, and am interested in how some results change if we work over fields of positive characteristic or fields that are ...
2
votes
2answers
56 views

The number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$

I am trying to find the number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$, where $\mathbb{Z}_3$ denotes the cyclic group of order $3$ and $D_{2n}$ denotes the dihedral group of order $2n$. ...
6
votes
1answer
75 views

Number of subgroups of an abelian p-group

Let $p$ be a prime number and let $n\in \mathbb{N}$. I know that every abelian group of order $p^n$ is uniquely a direct sum of cyclic groups of order $p^{\alpha_i}$ where $\sum \alpha_i = n$. Now the ...
2
votes
0answers
29 views

Size of conjugacy class of a quotient group

I am trying to recall what happened to the conjugacy class and centraliser when we quotient out by a normal subgroup. In particular, we know from Orbit-Stabiliser that $|G|=|ccl_G(g)||C_G(g)|$ for ...
0
votes
2answers
46 views

A proof about Automorphism in congruence class

Suppose $gcd(m,n)=1$, and let $F :Z_n→Z_n$ be defined by $F([a])=m[a]$. Prove that $F$ is an automorphism of the additive group $Z_n$. I find it is diffcult to prove $F$ is injective and surjective. ...
3
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0answers
38 views

Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
-2
votes
0answers
38 views

Number of conjugacy classes of a finite p-group [on hold]

Let $p\ne 2$ be a prime number and $G$ be a non-Abelian finite $p$-group. Is it true that the number of conjugacy classes of $G$ is not a power of $p$? I give you a background for this question: Lets ...
0
votes
0answers
26 views

Prove that the central product of $Z_4$ and $D_8$ is isomorphic to $Z_4$ and $Q_8$.

Let $Y_1=\{(1,1),(x^2,r^2)\},\;Y_2=\{(1,1),(x^2,-1)\}$, then prove $Z_4\times D_8/Y_1\cong Z_4\times Q_8/Y_2$. Since both of them have order $16$, it's not hard to list all of their elements, but is ...
-2
votes
1answer
21 views

Find all subgroup of $PSL(3,q)$ where is a power of an odd prime $p$ [on hold]

Let $G$ be a projective special linear group $PSL(3,q)$ where is a power of an prime $p$. Question : Find all subgroup of $G$.
2
votes
1answer
26 views

Normal subgroup with index that divides n!

Is this argument valid? If $G$ is a finite group with $n$ Sylow $p$-subgroups (in particular $n = 1$ mod $p$ and $n$ divides $|G|$), then $G$ permutes them acting by conjugation. Therefore there is a ...
3
votes
2answers
45 views

Do there exist (non-trivial) prime solutions to the equations $p^2 = 1$ mod $q$, $q = 1$ mod $p$?

Question: Do there exist odd primes $p$ and $q$ such that $$p^2 = 1 + qt,\quad q = 1 + ps$$ for some positive integers $s,t$? I've written some code which has verified that no solutions exist for $p,q ...
1
vote
0answers
31 views

Proof Verification: Classify all groups of order 6

I'm currently in an introductory group theory class, and I'm supposed to classify all groups of order 6 up to isomorphism without use of more advanced concepts like the Sylow theorems or even the ...
6
votes
1answer
527 views

Misunderstanding of Sylow theory

I think I have a misunderstanding a part of Sylow theory for groups or I have made a big mistake in my reasoning below. We have the following lemma in Sylow theory: Let $G$ be a finite group and ...
2
votes
1answer
44 views

How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$

How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$ This is my attempted proof. Proof: Any action of $\mathbb{Z}_3$ on the set $\{1, 2, 3, 4\}$ is equivalent to a homomorphism ...
0
votes
0answers
25 views

Examples for Converse of Lagrange's Theorem is false. [duplicate]

I know only one example , that is Alternating group of degree 4 , whose order is 12 , has no subgroup of order 6. Can you provide some other examples?
0
votes
0answers
12 views

Dualizing from co-invariants to invariants

I'm reading a paper where at some point the author is trying to gain some information about the dimension of some invariant space of some homology group of a space. For this, they prove the ...
0
votes
0answers
24 views

Bounding the exponent in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_0$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_1:=1+\operatorname{rad}(KG)$ is a p-group ...
-2
votes
0answers
46 views

Quaternion group of order 12 [closed]

What are the elements of the quaternion group of order 12 ? I found this on wikipedia but I don't know what are their elements.
0
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0answers
19 views

Deduce Schur-Zassenhaus theorem from Wedderburn-Malcev theorem?

Is it possible to derive Schur-Zassenhaus theorem (see e.g. https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem) from Wedderburn-Malcev theorem (see e.g. https://www.math.uni-bielefeld.de/~...
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votes
0answers
6 views

Isomorphism problem for the center of modular group algebras

Let $p$ be a prime number, $G,H$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. My ...
1
vote
0answers
6 views

A question to the ascending central chain in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. Let us focus on the sequence $G\cdot ...
0
votes
0answers
9 views

Defect of subnormality in modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $1+\operatorname{rad}(KG)$ is a p-group containing $G$. $G$ is ...
1
vote
1answer
25 views

Centralizers in p-groups

Let $G$ be a finite p-group and $g\in G$ such that $C_G(g)=C_G(g^p)$. Is it possible that for all $x\in G\setminus C_G(g)$ the identity $x^{\langle g^p \rangle}=x^{\langle g \rangle}$ is valid? E.g.,...
5
votes
0answers
53 views

Show that there is no simple group of order $3393$

Show that there is no simple group of order $3393$ The hint I was given was to look at Sylow $3$-subgroups. I know that $3393 = 3^2 \times 13 \times 29$ and that $G$ must contain a subgroup of order $...
2
votes
0answers
23 views

Proof that orthogonal conjugation is self-inverse

I am trying to prove that given a cyclic gropu $G$ and a subgroup $H$ then $(H^\bot)^\bot = H$, where $H^\bot = \{ \alpha \in G | \chi_\alpha(x) = 1 \forall x \in H\}$ and $\chi_\alpha$ are fourier ...
0
votes
1answer
35 views

Subgroup of $S_4$ generated by $\{(123), (12)(34)\}$

I refer to the following problem. Determine the subgroup of $S_4$ generated by $\{(123), (12)(34)\}$. In his solution to the problem the author makes the following claim: As $(123) \in A_4$ and ...
3
votes
1answer
41 views

Size of conjugacy class in subgroup compared to size of conjugacy class in group

Given: $\bullet$ A finite group $G$, an index 2 subgroup $H$, an element $a \in H$ $\bullet$ $[a]_H$ and $[a]_G$ are the conjugacy class in $H$ of $a$ and the conjugacy class in $G$ of $a$, ...
0
votes
0answers
40 views

A finite group of order $n$ has exponent $n$.

Definitions: The order $|G|$ of a group $G$ is the number of elements of $G$. The exponent of a group is an integer $n$ such that $x^n = e$ for all $x\in G$ ($e$ is the neutral element). ...
1
vote
1answer
31 views

Why is the number of Sylow 2 subgroups of simple group with order 60 not able to be 1 or 3?

I want to show that a simple group of order 60 is isomorphic to $A_5$. In the process, I am stuck at the part in which I have to show that the number of Sylow 2 subgroups (whose orders are 4) cannot ...
2
votes
3answers
109 views

What group is $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$?

I know that $\langle (1,1,1) \rangle$ in $\mathbb{Z}_5 \times \mathbb{Z}_4 \times \mathbb{Z}_8$ is isomorphic to $\mathbb{Z}_{40}$, but is there a way of writing what group it is (not what it's ...
1
vote
1answer
15 views

Show that $a$ generates the group of units $(\Bbb{Z}/F_n\Bbb{Z})^\times$

Let $F_n=2^{2^n}+1$ be a Fermat prime and let $a\in\Bbb{Z}$ such that $F_n\not| a$ and $a$ is not a quadratic residue modulo $F_n$. I want to show that the class $a+(F_n)$ generates the group of ...
1
vote
1answer
63 views

Power automorphism of elemantary abelian group

I proved that a subgroup A normalizes every subgroup of the minimal normal subgroup $N$, $N$ is an elementary abelian group of order $p^n$, $n>1$. It is clear that A induces a power automorphism ...
0
votes
0answers
43 views

No simple groups of given order.

I am trying to show the following: Prove that there are no simple groups of the given order: 42. 200. 231. 255. I understand that they need to be broken down into their prime factors. I was ...
0
votes
0answers
31 views

Equality of morphism of representations of $S_n$ and $A_n$

Let $S_n$ be the symmetric group and $A_n$ the Alternating group. Let $(V,\rho)$ be a simple representation of $S_n$. Let $(W,\pi)$ be a simple representation of $A_n$. Suppose $V=W$. I saw in some ...
4
votes
1answer
76 views

Is there a good way to show that the order of element in $S_7$ are at most $12$?

The only solution would be going through all cycle types of all permutations which is a lot of work. Is there any smarter solution than this one? Thank you in advance!
0
votes
0answers
41 views

On 1987, J.G.Thompson's conjecture

Let $G$ be a Group. The set of element orders of $G$ and the set of numbers of the same order elements in $G$ are denoted by $π_e(G)$ and $τ_e(G)$ (sometimes nse($G$)), respectively. Let $π(n)$ be the ...
-3
votes
0answers
25 views

A mapping problem.

To show onto. Choosen any element $c$ from $U(q')$, there exists a pre image in $U(q)$ such that the given relation holds. But I can't frame it out. Can anyone suggest how to proceed. Thanks in ...
3
votes
1answer
55 views

Finding the order of $\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle.$

Im new at abstract algebra stuff and im wondering whats the technique to prove this kind of stuff. Question: Let $G=\langle a,b | a^{8}=b^{2}=1, ab=ba^{3}\rangle$, prove that $|G|=16 $ and find all ...
2
votes
1answer
27 views

Focal subgroup is normal

When $H$ is a subgroup of $G$, we can define the focal subgroup of $H$ as $$H^\ast:=\langle h^{-1}h'\mid h,h'\in H, h'=h^g, g\in G\rangle.$$ I'm confused about the proposition "$H^\ast$ is a normal ...
4
votes
1answer
46 views

Characterizing group operation properties by its multiplication table

Let $G = \{x_1,\dots, x_n\}$ be a set equipped with an operation $*$. Let $A = [a_{ij}]$ be its multiplication table, $a_{ij} = x_i*x_j$. Assume $G$ has a identity $e$ (such that $e*x=x*e=x$ for all $...
0
votes
0answers
25 views

odd order group has non real irreducible representation

Let $G$ be a finite group such that $|G|$ is odd. show that G has a non real irreducible representation $\rho$ over $\mathbb{C}$. (in other words, is there a base $B$ of $V_{\rho}$ s.t. $[\rho(g)]_B \...
4
votes
2answers
176 views

What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
0
votes
0answers
26 views

Spin structure and characteristic classes

I do not know if anyone can help me with these doubts of spin structures and characteristic classes. 1) Is there an orientable manifold that is not spin? 2) Is there a finite group $ G $ such that ...
2
votes
1answer
39 views

Is there an easy way to determine the conjugacy classes of $SL(2,3)$?

I am trying to determine the conjugacy classes of $SL(2,3)$ (there are 7) and it is so long and boring, that I am starting to question all of my life choices leading up to this moment. I've read ...
2
votes
1answer
44 views

There is an element with order $p$ [duplicate]

Let $p$ is an odd prime and $G$ is a group which has $2p$ elements. Show that there exists at least one element with order $p$. I tried showing in 2 parts as $G$ is cyclic and not cyclic, but I ...
1
vote
0answers
24 views

Counting inversions of random elements in coxeter groups

I am trying to find a general interperetation to the following facts (pls be patient to read it). Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of ...
2
votes
0answers
68 views

$G$ group of order $n$. Can $[\operatorname{Sym}(G):\operatorname{Aut}(G)]$ and $|\operatorname{Out}(G)|$ be expressed in terms of $n$?

Let $G$ be a finite group, with $|G|=n$. $\mbox{¿}$Are there explicit expressions, in terms of $n$, for the two indexes: $k:=[\operatorname{Sym}(G):\operatorname{Aut}(G)]=k(n)$ $l:=[\operatorname{...
3
votes
1answer
18 views

Automorphisms of $D_4(q)$ (Chevalley group)

As a follow-up to my previous question, I need to investigate the action of certain outer automorphisms of $D_4(q)$ on specific $2$-subgroups (the defect groups of its blocks, if anyone is interested)....
2
votes
1answer
41 views

Either $H \triangleleft G$ or exist a conjugated subgroup $H^g \subseteq N_G (H)$, in which $g \in G$, with $H^g \neq H$.

Let $G$ be a $p-$group. If $H$ be a subgroup of $G$, prove that either $H \triangleleft G$ or exist a conjugated subgroup $H^g \subseteq N_G (H)$, in which $g \in G$, with $H^g \neq H$. In my opinion,...