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.

learn more… | top users | synonyms

0
votes
1answer
37 views

Semidirect Product Definition of Addition

Hello Everyone, I'm having a hard time trying to define the addition on this semidirect product, any suggestion would be appreciated. Thanks.
6
votes
0answers
64 views

Subgroup notation on Wikipedia

Surfing wikipedia about finite groups, I see a lot of notation similar to that found in this section (image below). Its meaning is not clear to me and I didn't find an explanation on wikipedia. Could ...
3
votes
2answers
27 views

Fundamental theorem of finitely generated abelian groups.

If $G :=\langle x,y,z \ | \ 2x+3y+5z = 0\rangle$ then find what group $G$ is isomorphic to. I think I'm supposed to use the fundamental theorem of finitely generated abelian groups, but I don't know ...
1
vote
0answers
42 views

Lucido's three prime lemma

I am looking for proof of this statement I encountered in a paper. $\textbf{(Lucido’s Three Primes Lemma)}$- Let $G$ be a finite solvable group. If $p, q, r $ are distinct primes dividing |$G$|, ...
1
vote
1answer
36 views

Given three non abelian group of order 8,two must be isomorphic.

True or false: Given three non abelian group of order $8$,two must be isomorphic. Solution: Theorem : A non-Abelian group of order $8$ is isomorphic either to $D_4$ or $Q_8$. I Think it is ...
2
votes
1answer
37 views

Group theory problem automorfism

Let $G$ be a finite group. If there exist an automorphism $f$ such that if $f(x)=x \iff x=e$ and $f(f(x))=x$ for all $x$ in $G$, then prove $G$ is Abelian.
2
votes
1answer
79 views

Generalization of a property Of $A_5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_5$ and $\langle H,K\rangle < A_5$. We can show that there exists a maximal subgroup $M$ of $A_5$ such that ...
1
vote
1answer
33 views

Eigenspace of finite abelian group

Let $\rho: G\to {\rm GL}_n(\mathbb{C})$ be faithfull representation of finite abelian group $G$ and $V$ is the eigenspace of some $g\in G$. Is it true that $V$ is also eigenspace for all $G$ (that ...
1
vote
2answers
34 views

If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then $T^n$ is the identity operator?

If $T$ be an invertible linear operator on a finite-dimensional vector space over a finite field , then is it true that $T^n = I$ ( the identity operator) for some positive integer $n$ ?
1
vote
1answer
34 views

How to check which subgroups of $D_4$ are normal

How do I check which subgroups of $D_4$ are normal? Trying all elements seems very cumbersome. So far, I know only basic theorems like Lagrange's and the homomorphism as well as the isomorphism ...
2
votes
1answer
45 views

quick way to prove $\mathbb{Z}_2 \times \mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$

I want to show that $\mathbb{Z}_2 \times \mathbb{Z}_3$ is isomorphic to $\mathbb{Z}_6$. The number of elements and their orders are equal but I don't see a way to prove that the groups are isomorphic ...
1
vote
1answer
41 views

cyclic group - automorphism - $f_a(x) = x^a$

Question: Let G be a cyclic group of order n. For each $a\in\Bbb{Z}$ define $f:G\to G ,f_a(x) = x^a$. Prove that $f_a$ is an automorphism of G if and only if $a$ is prime to $n$. I managed to show ...
2
votes
3answers
67 views

Homomorphisms into the General Linear Group

Let $G$ be a finite group of order $n \geq 2$. I want to prove that there always exists an injective homomorphism $\varphi:G \to GL_n(\mathbb R)$. Can you help?
3
votes
1answer
47 views

Conjugacy class size and simple group

For a finite group $G$, it is denoted by $N(G)$ the set of conjugacy class sizes. Let $G$ be a finite group and $H$ be a finite non-abelian simple group. Is it possible that $N(G)=N(H)$? In fact, I ...
3
votes
0answers
46 views

A topic for seminar in finite group theory.

I am doing a course on "Structure of finite groups" and we have a choice of giving a presentation on topics related to the course, Course outline is as follows- I was thinking of O-Nan Scott ...
1
vote
3answers
57 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
0
votes
0answers
21 views

showing that $G$ is super solvable.

every finite group that its order is square free ,have a non-trivial characteristic sylow subgroup and then it is super solvable. my problem is just in showing that it is super solvable, I want to ...
0
votes
0answers
15 views

algorithm to generate some type of words in a finite group

In order to solve a problem (that is off topic to this subject) I would like to know if there is an algorithm that gives me the following: Given a finite group $G$ and some of its elements e.g. $a,b,c ...
2
votes
1answer
49 views

For finite group $G$ when is $|Aut(G)| < |G|$?

If $G$ is a finite group then we know $|Aut(G)|$ divides $(|G|-1)!$ ; I want to ask , if $G$ is a finite group with more than one element then is it true that $|Aut(G)| < |G|$ ( I know it is ...
3
votes
1answer
37 views

Finding Sylow p-subgroups

I am trying some examples of finding Sylow p-subgroups in specific groups and looking for the most efficient way to do so. For example, lets say we need to find Sylow-3 subgroups in $A_4$ and $D_6$, ...
3
votes
1answer
28 views

Is it true that $n_p!\le |G|$?

Let $G$ be a finite group and $n_p:=|\text{Syl}_p(G)|$. Is it true that $n_p!\le |G|$ ? I've shown that it's true, but I'm not so sure, can you check my proof? Proof. Let $G$ act on ...
0
votes
1answer
37 views

Group Action and Orbits

I am looking at the following example which says find the orbit of $0$ under addition by $2$ and $3$ if $\mathbb{Z}_4$ acts on itself by addition. So to find the orbit of $0$ we are looking at the set ...
2
votes
2answers
45 views

Solution verification: $G$ and $G/H$ contain elements of same order

I just took my abstract algebra midterm, and was wondering if someone could confirm my solution to the following problem. Problem: Let $G$ be a finite group and let $H$ be a normal subgroup of ...
1
vote
1answer
30 views

Determine all the generators of $\mathbb{Z}_{25}^{\times}$

Determine all the generators of $\mathbb{Z}_{25}^{\times}$. Is there some way that I can use the fact that $\mathbb{Z}_{25}^{\times}$ is cyclic generated by $3$?
0
votes
0answers
24 views

Every finite group of square-free order is soluble

prove that every finite group which it's order is square free is soluble. I think it is enough to show that every sylow subgroup of this is cyclic. please tell me if my idea is right and if it is ...
-1
votes
1answer
24 views

centralizer of a chief factor

Let $G$ be a finite solvable group and $p$ be a prime. Let $G^*$ be the smallest normal subgroup of $G$ for which the corresponding factor is abelian of exponent dividing $p-1$. Show that every chief ...
1
vote
1answer
22 views

Question about Sylow subgroups

"Let $|G|=p_1^{e_1}\cdot \cdot \cdot p_t^{e_t}$ and let $G_{p_i}$ be the Sylow $p_i$-subgroup of $G$. The subgroup $S$ generated by all of the Sylow subgroups is $G$, for $p_i^{t_i}|~|S|$ for all ...
0
votes
0answers
6 views

Binary Tetrahedral Group

Consider the Binary Tetrahedral Group $2T.$ I am trying to prove that $Q_8$ is a subgroup of $2T$ using automorphisms. Is the full automorphism group of $2T$, $T$ where $T$ is the tetrahedral group? ...
1
vote
0answers
13 views

SO(3,q) to PGL(2,q)

Can anyone suggest a reference to an explicit formula giving, in the standard matrix notation, a homomorphism from SO(3,q) to PGL(2,q) (classical matrix groups: orthogonal of dimension 3 and ...
1
vote
1answer
22 views

Does this group theory question require an additional hypothesis?

The problem is to show that if $G$ is a finite group and for all nontrivial elements $a, b$ there exists an automorphism taking $a$ to $b$, then $G$ is a $C_p$ vector space, where $C_p$ is the group ...
1
vote
1answer
28 views

Calculation of polynomial in the finite field

I'm trying to understand the McEliece cryptosystem and I'm looking to this paper http://www.mif.vu.lt/~skersys/vsd/crypto_on_codes/goppamceliece.pdf On page 26 they are calculating syndrome and ...
0
votes
0answers
13 views

centre of an infinite p group may be trivial. [duplicate]

I have to prove this and can not seem to find any such example. I know for finite groups, centre is always non trivial, but what example is there for infinite. Thanks
1
vote
2answers
48 views

A question in Abelian Group

Suppose $G$ is a finite Abelian group, and $U$ is a cyclic subgroup of $G$ with the maximal order. Choose $y$ in $G-U$ such that $|y|$, the order of y, is the minimal. Suppose $p$ is a prime number ...
2
votes
2answers
43 views

Commutativity of special type of permutations

Let $p_1,p_2$ be permutations on the set $S=\{1,2,...,n \}$ and define $U_1:=\{k\in S:p_1(k)\ne k\}$ , similarly we can define $U_2$ . Now I can prove that if $U_1 \cap U_2=\phi$ , then ...
1
vote
0answers
26 views

Is membership in the index 2 subgroup of $Sp_4(\mathbb{F}_2)$ detected by a polynomial in the matrix entries?

I learned from Magma that $Sp_4(\mathbb{F}_2)$ has an index-2 subgroup isomorphic to $A_6$. Is it possible, given a matrix $M\in Sp_4(\mathbb{F}_2)$, to detect membership in this subgroup using a ...
1
vote
2answers
34 views

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true?

Let $p,q$ are distinct primes and $G$ be a group of order $pq$ then which of the following is true? $1.G$ has exactly $4$ subgroups upto isomorphism. $2.G$ is abelian. $3.G$ is isomorphiq to a ...
0
votes
2answers
71 views

Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
2
votes
1answer
21 views

Let $S$ be a subset of $R$ such that we have associative relation $*$ defined on $S\times S\to S$ with some properties…

Let $S$ be a subset of $\\R$ such that we have associative relation $\\*\\$ defined on $S\times S\to S$ with $$a*b*a=b \hspace{0.5cm} \forall a,b\in S, \hspace{1.5cm} \exists e \hspace{0.5cm} ...
1
vote
1answer
35 views

Why in this sense this homomorphism is injective?

In this proof:enter link description here Page 19, it gives a construction of outer automorphism of $S_6$,it sends $S_6$ to Perm($S_6$/H)= $S_6$, and by the former proof, it is injective.However, it ...
4
votes
2answers
55 views

Group with $p+1$ Sylow $p$-subgroups

Given a group $G$ with $p+1$ Sylow $p$-subgroups, I've deduced that $R = P \cap P'$, where $P, P'$ are Sylow $p$-subgroups, has index $p$ in each of $P, P'$; and that all $p+1$ Sylow $p$-subgroups of ...
0
votes
0answers
97 views

finite solvable group with a certain property

Let $G$ be a finite solvable group and for each proper normal subgroup $N$ of $G$, $\frac{G}{G^{\prime}N}\cong \Bbb{Z}_p\times\Bbb{Z}_p$ or $\Bbb{Z}_{p^n}$, where $n\geq 1$, $p$ is a prime number ...
1
vote
1answer
45 views

properties on groups of order $p^2qr$

I read somewhere that if $|G|=p^2qr$, $H\subseteq G: |H|= p^2q$, $p>q>r$ primes, then if only $H$ is maximal subgroup, then $H$ is Abelian. Is this problem correct? Are there any same properties ...
0
votes
1answer
15 views

Primary decomposition of $Z_{1001}$ as a group of multiplication

The question is asking for the primary decomposition of $Z_{1001}$ as an abelian group under multiplication. So I did the following. By Euler $\phi$ function, I count the number of integers ...
0
votes
1answer
13 views

Using Lattice Isomorphism Theorem

I am working on this for my algebra class and I am stuck at the very end. $\textbf{QUESTION:}$ Let $p$ be a prime and let $G$ be a group of order $p^\alpha$. Prove that $G$ has a subgroup of order ...
3
votes
1answer
50 views

No simple groups of order 9555: proof

While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says: "Moreover, since 7 does not ...
1
vote
1answer
29 views

About primary decomposition of $\mathbb{Z}_7$

I want to find the primary decomposition of $\mathbb{Z}_7^*$ under multiplication group action. So I know 7 is a prime number so it is a group of order 6. So possibilities are $\mathbb{Z}_7^*\cong ...
1
vote
1answer
56 views

Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
0
votes
0answers
26 views

Describe all subgroups of $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$

Is it possible to give a general discription of all subgroups of the group $\operatorname{Inn}(\operatorname{GL}_2(\mathbb F_{p^n}))$ of inner automorphisms of $\operatorname{GL}_2(\mathbb F_{p^n})$? ...
-1
votes
1answer
31 views

Subgroups in $G$ of the form $gHg^{-1}$

Let $G$ be a group and $H$ be a subgroup of finite index. Prove that there is only a finite number of distinct subgroups in $G$ of the form $gHg^{-1}$ where $g$ belongs to $G$.
2
votes
1answer
28 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...