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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7
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3answers
235 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
107 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
69 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
37 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 ...
3
votes
1answer
186 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
88 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
87 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
87 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
80 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
160 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
84 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!
2
votes
1answer
148 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
149 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
129 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 ...
3
votes
1answer
95 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
351 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. ...
7
votes
1answer
152 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
120 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
60 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
209 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
34 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
151 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
51 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
108 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
41 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
155 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
76 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
37 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 ...
3
votes
1answer
112 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
74 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
235 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
204 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
63 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
107 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
163 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
164 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
241 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
65 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
388 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 ...
0
votes
1answer
63 views

prove that the order of the elements of restricted direct product is finite .

if $G_i = \Bbb Z/p_i \Bbb Z$ ($\Bbb Z$ means integers) where $p_i$ is the ith integer prime , I=the positive integers show that , every element of the restricted direct product of the $G_i$'s has ...
1
vote
1answer
189 views

Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible ...
5
votes
1answer
86 views

Regarding subgroups of $GL_{2}(\mathbb{F}_q)$ of order $p$, where $p$ is an odd prime dividing $q+1$.

The problem (this is not a homework problem, though it does feel like one): Suppose p and q are primes such that $p \vert q+1$ and p is odd. Then all subgroups of $GL_{2}(\mathbb{F}_{q})$ of order p ...
1
vote
2answers
52 views

Number of abelian groups Vs Number of non-abelian groups

I would like to see a table that shows the number of non-abelian group for every order n. It is a preferable if the table contains the number of abelian groups of order n (this is not necessary ...
3
votes
1answer
272 views

Classify all groups of order 182

In studying for an upcoming prelim, I came across this problem: Classify all groups of order $182 = 2*7*13$. Now, the standard tricks here are to look at Sylow's theorems or semi-direct ...
3
votes
1answer
109 views

Find all irreducible representations of $A_4$ over $\mathbb{F}_p$ where p is an odd prime

I have no issue with finding representations over $\mathbb{C}$ but I'm not quite sure how to find them over finite fields, particularly when that field might not be algebraically closed or the ...
1
vote
1answer
84 views

Show that there are $36$ $5$-Sylow subgroups

Given the symmetric group $S_6$, we consider a $5$-Sylow subgroup. How can one show that this subgroup is isomorphic to $\mathbb{Z}/ 5 \mathbb{Z}$? I have to show that there exist $36$ such ...
7
votes
1answer
220 views

Certain Sums of Conjugacy Class Sizes of Symmetric Groups

Suppose $n$ is a natural number and $\lambda$ is an unordered integer partition of $n$ such that $\lambda$ has $a_{\lambda,j}$ parts of size $j$ for each $j$... Let $c_\lambda$ be the conjugacy class ...
1
vote
1answer
92 views

faithful irreducible representation of $A_{4} \times Q_{8}$

Construct a faithful irreducible representation of the group $A_4 \times Q_8$ $A_{4}$ is the alternating group $Q_{8}$ is the quaternions
1
vote
1answer
108 views

If $V$ is an irreducible representation then is $S(V\otimes V)$?

Let $V$ be an irreducible $FG$-module of dimension $2$. Is $S(V\otimes V)$ irreducible? Why? $G$ is a finite group. $F$ is a field, its order is unspecified. $S(V\otimes V)=\{x \in V\otimes V : ...
2
votes
1answer
96 views

$G$ is semiregular implies its centralizer is transitive

How do I prove that the centralizer of every semiregular group is transitive? This is Exercise 4.5 in [Wielandt, Finite Permutation Groups]. Recall that a permutation group $G \le S^\Omega$ is ...