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
votes
2answers
79 views

Can we get the line graph of the $3D$ cube as a Cayley graph?

Given a graph $G=(V,E)$, the line graph of $G$ is a graph $\Gamma$ whose vertices are $E$ (the edges of $G$) and in $\Gamma$, two vertices $e_1,e_2$ are connected if, as edges in $G$, they share an ...
1
vote
0answers
84 views

Constructing automorphisms

Given the nonabelian group $G$ of order $p^3$ where $p$ is a prime satisfying $p\equiv1\pmod4$, that is $G=\mathbb{Z}_p^2\rtimes\mathbb{Z}_p$ or $\mathbb{Z}_{p^2}\rtimes\mathbb{Z}_p$. Let $H$ be a ...
1
vote
1answer
83 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
2
votes
1answer
84 views

Schmidt group and sylow subgroup

Let $G$ be Schmidt group. If $Q=\langle a\rangle$ is $q$-subgroup of $G$, then $a^q \in Z(G)$
0
votes
2answers
93 views

how the set of all inner automorphisms of each group $G$?

1] show the set of all inner automorphisms of each group $G$ is a normal subgroup of the group of all automorphisms $G$?
3
votes
1answer
64 views

Is meta-$p$-nilpotent for all $p$ the same as meta-nilpotent?

I've been working on understanding solvable groups one prime at a time, and it has been very helpful. Perhaps this question is enjoyable to more than just me: Definition: A finite group is said to be ...
1
vote
2answers
105 views

Metanilpotent groups and saturated formation

Class of all metanilpotent groups is a saturated formation ? How do I prove
3
votes
2answers
335 views

Index of maximal subgroups of $p$-solvable groups

If $G$ is finite $p$-solvable group (every chief factor has order either a power $p$ or relatively prime to $p$) must every maximal subgroup have index either a power of $p$ or relatively prime to ...
2
votes
3answers
1k views

How to determine the number of isomorphism types of groups of a given order?

if $G$ is a group whose order is $n$ can we determine the number of isomorphism types for this number or not ? for instance, if $n=4$ we have 2 types, $Z_4$ and $Z_2 \times Z_2$ " Klein 4-group" ...
2
votes
1answer
123 views

Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.

Prove $D_{8n} \not\cong D_{4n} \times Z_2$. My trial: I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows: Suppose $D_{4n}$ is isomorphic ...
0
votes
2answers
81 views

finite group and $M$ is a maximal subgroup

Let $G$ be a finite group and $M$ is a maximal subgroup of $G$. Prove $\forall g \in G$, $M^{g}$ is a maximal subgroup of $G$.
10
votes
3answers
283 views

Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$

If the order of $G$ is $p^2$ then how do I show that $G$ is isomorphic to $\mathbb Z_{p^2}$ or $\mathbb Z_p\times\mathbb Z_p$.
1
vote
1answer
127 views

$G$ soluble and Unique minimal normal subgroup

Let $G$ be a soluble group and $N$ is only minimal normal subgroup of $G$. Is this $N=C‎_{G}‎(N)$ true?
0
votes
1answer
83 views

The symmetric group $S_4$ and cycles as product of permutations

i have a question, i think it is very easy but i didn't see it: I know that: $$S_4=\langle(1~~2),(2~~3),(3~~4)\rangle$$ Now i want to write $(1~~2~~3)$ and $(1~~2~~3~~4)$ as products of them. ...
3
votes
1answer
46 views

Trace of the action of the Hecke algebra

Let $G$ be any finite group, $H$ a subgroup of $G$, and $\mathcal{R}$ the Hecke algebra associated to this data (i.e. the space of $H$-bi-invariant maps $G \longrightarrow \mathbb{C}$ with the ...
4
votes
1answer
310 views

Can a group of order $55$ have exactly $20$ elements of order $11$? (Clarification)

After I asked this question, which I now understand, I came across a similar question. But I don't understand the answer that was chosen; particularly the part about $34$ elements of order $5$ (but ...
6
votes
1answer
100 views

Isomorphism of complements in semi-direct products

Suppose $G$ is a finite group with normal subgroups $M,N$ and subgroups $H,K$ such that $M \cong N$, $MH=NK=G$, and $M \cap H = N \cap K = 1$. Is it the case that $H \cong K$? Clearly $H \cong G/M$ ...
6
votes
1answer
93 views

Normal products and radicals in finite groups

If $G$ is a finite group with normal subgroups $M$ and $N$, then $MN$ is a subgroup, called the normal product of $M$ and $N$. If $\mathcal{F}$ is a set of finite groups closed under isomorphism and ...
2
votes
0answers
96 views

Schmidt group and maximal subgroups

Let $G$ be a Schmidt group, a minimal non-nilpotent group, so that $G$ is not nilpotent, but every proper subgroup of $G$ is nilpotent. I want to prove $G$ has precisely two classes of maximal ...
3
votes
1answer
99 views

Does the class of soluble groups whose $p$-length is $\leq 1$ for all $p$ form a saturated Fitting formation?

Let $\mathcal{F}$ be the class of all soluble groups $G$ such that the $p$-length $l_{p}(G) \leq 1$ for all primes $p$. How do I show ‎$\mathcal{F}$ is a saturated Fitting formation?
0
votes
2answers
192 views

On techniques of using Sylow Theorems to show that groups of certain orders are not simple

As seen in this answer, a group of order 144 is not simple. Now, I understand the main part of the answer, i.e. where it is concluded that, upon deducing that $n_3 = 16$, it is forced that $n_2 = 1$, ...
2
votes
2answers
103 views

Minimal subgroups lie in the center so group is nilpotent

Let $G$ be a group of odd order. If every minimal subgroup lies in the center, prove that $G$ is nilpotent . Thanks!
3
votes
1answer
234 views

The Fitting subgroup centralizes minimal normal subgroups in finite groups

Let $G$ be a finite group: If $N$ is a minimal normal subgroup of $G$, then $F(G) \leq C_G(N)$. Here $C_G(N)$ denotes the centralizer of $N$ in $G$, and $F(G)$ denotes the Fitting subgroup of $G$.
1
vote
1answer
173 views

How to define an automorphism for $S_3 \times C_8$ in GAP?

Consider the group $W:= S_3 \times C_8$. How can I define an automorphism for $W$? For example $f:W ‎\longrightarrow‎ W$; $f(x,y)=(x,y^{5}g(x))$ where $g:S_3 ‎\longrightarrow‎ C_2$ is defined as ...
7
votes
1answer
139 views

Subgroup of elements of order at most $2^{m}$

The problem A5 in Putnam 2009 reads as follows: Is there a finite abelian group $G$ such that the product of the orders of all its elements is $2^{2009}$? The answer is No. I am reading the ...
4
votes
1answer
99 views

What can we say about the order of a group given the order of two elements?

If I know that a group of finite order has two elements $a$ and $b$ s.t. their orders are $6$ and $10$, respectively. What statements can be made regarding the order of the group? I know by ...
7
votes
2answers
553 views

number of subgroups index p equals number of subgroups order p

I'm doing an exercise in Dummit book "Abstract Algebra" and stuck for a long time. I think I'm doing in the right way but I can't finish it. Hope someone can help me. I really appreciate it. ...
8
votes
1answer
207 views

Are there any distinct finite simple groups with the same order?

In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two ...
2
votes
1answer
167 views

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does ...
7
votes
0answers
68 views

Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?

If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ ...
2
votes
0answers
305 views

Characters of double groups

Problem: I want to do some calculations with the character projection operator to investigate the irreducible representations of wave functions. Until now, I did these calculations for simple ...
1
vote
1answer
37 views

Show the $\mathbb{C} S_3$-module of dimension 2 has $S(V \otimes V)$ is not irreducible

Consider the $\mathbb{C} S_3$-module of dimension 2, call it $V$. I want to concretely show that $S(V \otimes V)$ is not irreducible. I found a representation for $S_3$ over $\mathbb{C}$ of degree ...
6
votes
2answers
155 views

How does this strange phenomena happen in quotient of groups ?

in my question , here , i learned a strange fact from the comments which was a surprise for me on the answer of landsacpe ! and this surprise it : if $G$ is a group , $H$ and $K$ are two normal ...
1
vote
1answer
56 views

How can i create a presentation of a group ?

in Dummit and Foote , the notion of presentation is introduced in section 1.2 which talks about dihedrial group of order $2n$. and after this , it was rare to talks about presentation throw the ...
4
votes
3answers
125 views

Extending and inducing irreducible representations

I sense this may be a simple question, but it is one I haven't been able to find an answer for, possibly due to the use of different terminology. Referring to this question: Faithful irreducible ...
2
votes
1answer
62 views

What is the smallest dimension possible for a representation of $D_8 \times Q_8$ which is faithful over $F$?

Consider $D_8 \times Q_8$, where $D_8$ is the dihedral group of order 8; $Q_8$ the quaternions. Let $F$ be a field of characteristic not equal to 2. What is the smallest dimension possible for a ...
-1
votes
1answer
160 views

prove that , there is no element $a , b$ of the group $G$ which satisfy this property

let $G=(x) \times (y) $ where $(t)$ is the group generated by $t$ , $|x|= 8 , |y|=4$ let $H=(x^2y , y^2 )$ be isomorphic to $Z_4 \times Z_2 $ prove that , there are no elements a,b of G such that ...
1
vote
1answer
92 views

Compute the order of the following elements in a group

Let $G = \mathbb Z_{84}$. Let $g,h \in G$, with $g = 6, h = 80$. Compute $|g|, |h|$ and $|gh^{-1}|$.
2
votes
1answer
38 views

A question of finite Group [duplicate]

Let $G = A_5$, the alternating group of degree five,. Let $\pi = \{2,3\}.$ Prove that $M$ is a maximal $\pi$-subgroup of $G$ if, and only if, $M\cong A_4$ or $M\cong S_3$, where $A_4$ is the ...
4
votes
2answers
150 views

Probabilistic Interpretation of Burnside's Lemma

Burnside's Lemma states that $N$, the number of orbits when a group $G$ acts on a set $X$ is given by $$N = \frac{1}{|G|} \sum_{g \in G} |\text{Fix } g|$$ The standard proof involves applying the ...
2
votes
1answer
75 views

How to find the order of $ X_k$?

Let $G_k = \Bbb Z_3 × · · · × \Bbb Z_3$. Let$ \,\,\alpha(z_1, . . . , z_{k−1}, z_k )=(−z_1, . . . ,−z_{k−1}, z_k ) \text{ where} \,\,z_i \in\Bbb Z_3$ for $i = 1, 2, . . . ,k$. Then $α ∈ ...
12
votes
2answers
278 views

$G$ abelian group with $\vert G\vert =mn$ and $\gcd(m,n)=1$.

I hope this is not a duplicate. First of all, in what follows I'm not allowed (unfortunately) to use the structure theorem for abelian groups. I'm asked to prove the following: Let $G$ be an ...
6
votes
2answers
286 views

Elementary Abelian $p$-Subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number. If $G$ is a finite group, an elementary abelian $p$-subgroup of $G$ of rank $r$ is any group $E\subset G$ such that $E\cong (\mathbb{Z}/p)^{\oplus r}.$ Let $\mathcal{E}_r$ ...
3
votes
3answers
70 views

Existence a certain subgroup of a group

‎Let ‎‎$G$ ‎be a‎ ‎finite group ‎such ‎that ‎‎$G=P\rtimes Q‎‎‎‎$ ‎where ‎‎$P\in {\rm Syl}_p(G)‎‎$;‎ ‎‎$‎‎P\cong \Bbb{Z}_p\times \Bbb{Z}_p‎$ ‎‎‎‎ and ‎$Q\in {\rm Syl}_q(G)$; ‎$‎‎|Q|=q$ (‎$‎‎p, q$ ‎are ...
4
votes
1answer
128 views

Element of order $2n$ in symmetric group $S_n$

I've been recently reading some articles about orders of elements in $S_n$ and I know that in order to find max order in $S_n$ we can use Landau function though I think that for small $n$ it is better ...
2
votes
1answer
283 views

Relationship between number of conjugacy classes and number of irreducible representations of a group

For a finite group G the number of irreducible representations over an algebraically closed field F is at most the number of conjugacy classes whose sizes are coprime to the characteristic of F. What ...
8
votes
1answer
217 views

Applications of computation on very large groups

I have been studying computational group theory and I am reading and trying to implement these algorithms. But what that is actually bothering me is, what is the practical advantage of computing all ...
3
votes
1answer
366 views

Degrees of faithful irreducible representations of $\mathbb{Z}_n$ over finite fields

I came across the following theorem while studying representation theory over finite fields. A cyclic group $\mathbb{Z}_n$ has a faithful irreducible representation of degree $d$ over $\mathbb{F}_p$ ...
1
vote
1answer
75 views

A condition for a finite group $G$ be nilpotent

Is true that a finite group $G$ is nilpotent if and only if $[x,y]=1$ for all $x,y \in G$, such that $(\mid x\mid, \mid y \mid) = 1$, where $[x,y] = x^{-1}y^{-1}xy$, ie, is the commutator of $x$ and ...
5
votes
2answers
564 views

Proof of Fundamental Theorem of Finite Abelian Groups?

The only proofs I've seen of this tend to involve a few intermediate results and a couple of induction proofs with some clever constructions in them. They aren't hard to follow and they're pretty ...