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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.

4
votes
0answers
24 views

Do there exist two finite groups $H$ and $K$, satisfying specific conditions?

Let’s define $\sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $\sigma(H) = |H| + |K| = \sigma(K)$, $|H|$ is even ...
5
votes
1answer
44 views

$H \leq \mathbb{Z}_q^n$ and $H \cong \mathbb{Z}_q^m$ implies that $\mathbb{Z}_q^n / H \cong \mathbb{Z}_q^{n-m}$

Given a pair of positive integers $n,q$ and a subgroup $H \leq \mathbb{Z}_q^n$ such that $H \cong \mathbb{Z}_q^m$ for a positive integer $m < n$ then show that $$ \mathbb{Z}_q^n / H \cong \...
5
votes
1answer
63 views

Do there exist finite non-cyclic groups $H$ and $K$, satisfying the specific condition?

Let’s define $\sigma(G)$ as the sum of orders of all normal subgroups of a finite group $G$. Do there exist two finite groups $H$ and $K$ such, that $\sigma(H) = |H| + |K| = \sigma(K)$ and $H$ is non-...
20
votes
2answers
558 views

A finite group such that every element is conjugate to its square is trivial

Suppose $G$ is a finite group such that $g$ is conjugate to $g^2$ for every $g\in G$. Here's a proof that $G$ is trivial. First, observe that if $\lvert G\rvert$ is even, then $G$ contains an ...
1
vote
1answer
24 views

There is no generating set of size $|S| \le \log_2|G|$

Let $m\in \mathbb{N}$. Show that there exists a finite group $G$ with $|G|\gt m$ such that every subset $S$ of $G$ with $|S| \lt \log_2|G|$ is not a a generating set of $G$. I find this question ...
3
votes
1answer
44 views

Non cyclic group of order $8$ having exactly one element of order $2$

Let $G$ be a non-cyclic group of order $8$ having exactly one element of order $2$. Prove that $G$ is generated by elements $a$ and $b$ subject to the relations $a^4=1$ and $a^2=b^2$. I can start ...
0
votes
0answers
24 views

Show that there exists a non-trivial automorphism on $G$. [duplicate]

Let $G$ be a finite group with at least four elements such that $g^2=e$ for all $g \in G$. Show that there is a non-trivial automorphism on $G$. How do I proceed to prove it? Please help me in ...
0
votes
2answers
39 views

Why is 9 a generator of a group Z28?

It says in my book that generators are relatively prime to $28$, so that would be a set of $\{1,3,5,9,11,13,15,17,19,23,25,27\}$, ok i get that. But why is $9$ and in there? I can express $9$ as $3^2$,...
0
votes
2answers
36 views

About the set of periods of a group subset [on hold]

Let $G$ be a group and $A\subseteq G$. Put $$ S(A):=\{g\in G: gA=A\}. $$ It can be shown that $S(A)$ is a subgroup of $G$, and $S(A)=A \ \iff \ A\leq G$. Now, is it true that: (1) $|S(A)|\leq |...
3
votes
1answer
39 views

Does there exist a nearly immaculate group not of the form $C_{2^n}$?

Define a nearly immaculate group as a finite group $G$, such, that the sum of the orders of all its normal subgroups is $2|G| - 1$. It is quite obvious to see, that all groups of the form $C_{2^n}$ ...
0
votes
1answer
46 views

How to represent the groups symbolically?

What will be the group if the groups given by following descriptions were represented using symbols/notation (like $Z_p$ etc.) 1). Cyclic group of order $p^2 q$. 2). Semidirect product of cyclic ...
1
vote
3answers
53 views

How many elements of order $n$ does $\bigcup_n^\infty\{e^{\frac{2ik\pi}{n}}:0\leq k\leq n-1\}$ have?

I came across a question which I though I managed to solve. After looking at the official solution for it I got very confused. The Question $H=\bigcup_{n=2}^\infty\{e^{\frac{2ik\pi}{n}}:0\leq k\leq ...
-2
votes
1answer
45 views

Solvable, non-nilpotent group with nilpotent commutator subgroup

What is the smallest example of a finite solvable, non-nilpotent group $G$, such that its derived subgroup $G'$ is nilpotent, but not abelian?
-1
votes
1answer
27 views

Group of even order contains an element of order $2$ (Explanation) [duplicate]

I am working on the problem: If G is a group of order 2n, show that the number of elements of G of order 2 is odd. which already has solution (in the link below), please explain the solution in ...
6
votes
0answers
133 views
+50

Irreducible 2-Brauer characters of $S_5$

Beginning with the ordinary character table of the symmetric group $S_5$, one immediately gets the following Brauer characters in characteristic two: $\begin{array}{c|c|c|c} S_5 & () & (...
2
votes
3answers
122 views

Understanding Converse of Lagrange's theorem!

Lagrange's theorem states that for any finite group $G$, the order (number of elements) of every subgroup $H$ of $G$ divides the order of $G$ .......(1) The converse of Lagrange's theorem is if $x$ ...
6
votes
1answer
38 views

What is the maximal $m$, such that $\mathbb{Z}_2^m \leq GL(n, 2)$?

Is there any closed formula for the function $m(n)$, that is defined as the maximal $m$, such that there is $GL(n, 2)$ has a subgroup isomorphic to $\mathbb{Z}_2^m$? The only things I know currently, ...
1
vote
1answer
41 views

Problem from Herstein's Topics in Algebra

Let $G$ be a group in which, for some integer $n > 1$, $(ab)^n = a^nb^n$ for all $a, b ∈ G$. Show that (a) $G(n) = \{x^n | x ∈ G\}$ is a normal subgroup of $G$. (b) $G(n- 1) = \{x^{(n- 1)} | x ∈ G\}...
0
votes
1answer
50 views

A finite group of even order, whose $2$-Sylow subgroups are cyclic, is not simple. [duplicate]

Let $G$ be a finite group of even order, whose $2$-Sylow subgroups are cyclic. Show that $G$ is not simple. I was trying to use Cayley theoem, place $G$ in some $S_m$ and get a contradiction with $...
0
votes
0answers
28 views

A Necessary and Sufficient Condition to Ensure a Finite Abelian Group Is Cyclic [duplicate]

Consider a finite abelian group $G,$ we have that $G$ is cyclic if and only if for each positive integer $n,$ the set $\{x \in G \,|\, x^n = e_G \}$ has at most $n$ elements. First, let us assume ...
0
votes
0answers
25 views

Group of rectangular Young Diagrams

Consider set $A$ of all Young diagrams in rectangle $m\times n$. Is there a way to define dot operation $A\times A \to A$, so $(A, \cdot)$ would be a group (preferably Abelian)?
2
votes
1answer
40 views

Subgroups of $SL(2,q)$

I need a list of all subgroups of $SL(2,q)$. When $q$ is even, $SL(2,q) \cong PSL(2,q)$ and in Dickson's "Linear Groups" there is a list of all subgroups of $PSL(2,q)$. But where can i find a list of ...
6
votes
2answers
85 views

Is there a way to describe all finite groups $G$, such that $Aut(G) = S_3$?

Is there a way to describe all finite groups $G$, such that $Aut(G) = S_3$? Two groups, that definitely satisfy that condition are $S_3$ itself (as it is a complete group) and $\mathbb{Z}_2 \times \...
4
votes
3answers
306 views

Do a subgroup and quotient determine the original group?

I am wondering whether there is a counterexample which shows that subgroups and quotients don't determine the group. More precisely, suppose there are two groups $G_1, G_2$ such that all of their ...
3
votes
1answer
28 views

A minimal normal subgroup of a $p$-solvable group is either a $p$-group or $p'$-group.

A minimal normal subgroup of a $p$-solvable group is either a $p$-group or $p'$-group. Let $G$ be a finite group and $p$ be a prime. Then $G$ is $p$-solvable if (i) $G$ has a subnormal series where ...
0
votes
0answers
27 views

Representation of finite groups up to isomorphism

Q1) If it is mentioned some thing like "a group $G$ consists of $p^2$ conjugate subgroups of order $q$" (as an example), is there a possible way to represent $G$ as isomorphic to some group by using ...
4
votes
0answers
78 views

Is there a notion “above” that of perfect numbers?

When trying to understand a notion, it often gives great insight to see it as the "shadow" of something bigger, carrying more information. The notion of categorification relies on this idea. A basic ...
4
votes
1answer
92 views

Classify all groups of order $4165$

Classify all groups of order $4165=5(7^2)17$. I've determined the following possibilities for each of the sylow subgroups $r_5 = 1$ $r_7 = 1$ or $5(17)$ $r_{17} = 1$ or $5(7)$ I'm trying to show ...
4
votes
2answers
76 views

Solutions of the equation $(m! + 2)\sigma(n) = 2n \cdot m!$ where $5 \leq m$

Are there any pairs of natural numbers $(m, n)$, with $5 \leq m$, other than $(5, 15128)$ and $(6, 366776)$, that satisfy the condition $(m! + 2)\sigma(n) = 2n \cdot m!$, where $\sigma(n)$ denotes the ...
3
votes
1answer
57 views

Find maximal permutation group $G$ such that a polynomial is $G$-invariant

I don't know if this is a trivial question. But because I lack some background I would need advise or a reference. I have an $n$-variate polynomial over $\mathbb Q$, say $f$, and I am interested in ...
0
votes
1answer
32 views

Quotient by kernel of a character

Let $G$ be a finite group, and $\chi$ a linear character of $G$. I've read that the quotient $G/ker(\chi)$ is cyclic. Why is this true? I also wonder about the quotient if $\chi$ is not linear.
-4
votes
0answers
55 views

Let $G$ be a group from order $2^n$ defined by: $G=\langle a,b: a^{2^{n-2}}=b^2=(ab)^2\rangle$ .

Let $G$ be a group from order $2^n$ defined by: $G=\langle a,b: a^{2^{n-2}}=b^2=(ab)^2\rangle$. find all the subgroups of $G$. I know that the order of the subgroups of G dividing the order of G ...
5
votes
2answers
190 views

Any finite group is a subgroup of an orthogonal group [duplicate]

Prove that any finite group of order $n$ is isomorphic to a subgroup of $\mathbb{O}(n)$, the group of $n\times n$ orthogonal real matrices. Attempt: Let $G$ be a group of order $n$. Then $G$ is ...
8
votes
7answers
153 views

Conditions for cyclic quotient group

Let $G$ be an arbitrary finite group and $H$ a normal subgroup. What are some good conditions on $H$ that make the quotient $G/H$ cyclic? I want to avoid any further restriction on $G$.
2
votes
1answer
52 views

$H,K \lt G$ with $HK=G$

$G$ is a group and $H \lt G$ and $K \lt G$, with $|G|=n$ then If $GCD( (G:H) ,(G:K))=1$ then $G=HK$ Any thoughts? Edit by Batominovski: To prevent this thread from being closed or getting ...
0
votes
0answers
26 views

Let $G=\langle (1,2,3,4,5,6,7), (2,4,3,7,5,6) \rangle$

Let $G=\langle (1,2,3,4,5,6,7), (2,4,3,7,5,6) \rangle$. I need to find sylow basis of $G$. since the order of $(1,2,3,4,5,6,7)$ is 7 so we can prove that $n_7=1$ then it means that $\langle (1,2,3,...
-1
votes
0answers
68 views

Let $G$ be a group from order $2^n$ defined by: $G=\langle a,b: a^{2^{n-2}}=b^2=(ab)^2\rangle$.

Let $G$ be a group from order $2^n$ defined by: $G=\langle a,b: a^{2^{n-2}}=b^2=(ab)^2\rangle$. find all the conjugacy classes of $G$. I found: $\{a^i, a^{-i}\} \ (for\ 0<i<2^{n-2}) $ $\{a^...
1
vote
1answer
39 views

Find two finite groups $G_1$ and $G_2$ such as:

Find two finite groups $G_1$ and $G_2$ such as: $1)$ $|G_1|=|G_2|$ $2)$ for all prime $p$ every $p$-sylow subgroup of $G_1$ isomorpic to $p$-sylow subgroup of $G_2$. but $G_1\not\cong G_2$. hello ...
2
votes
1answer
59 views

Finite abelian groups whose subgroups of the same order are all isomorphic

I am trying to find describe a finite abelian group whose subgroups of the same order are isomorphic. That is to say, if $H$ and $K$ are two subgroups of $G$ with $|H|=|K|$, then $H\cong K$. My ...
5
votes
1answer
71 views

Classifying groups of order 585

I am trying to classify the groups of order 585. (It is known that there are 4 of distinct non-isomorphic groups, but I am not assuming it.) The question further asks to show that any group of this ...
0
votes
0answers
65 views

Is $(ab)^n=a^nb^n$ condition on finite group implies that group to be abelian?

I was encountered following problem : Let G be the finite abelian group with order not divisible by 3 also $\forall a,b\in G$ such that $(ab)^3=a^3b^3$ then G is abelian. This I am able to prove using ...
4
votes
1answer
48 views

Example of a non-nilpotent finite group $G$ so that every non-trivial normal subgroup of $G$ intersects $Z(G)$ non-trivially?

It is well-known that if $G$ is a nilpotent group, then every non-trivial normal subgroup of $G$ has non-trivial intersection with $Z(G)$. I would like to find examples of non-nilpotent finite groups ...
13
votes
3answers
190 views

Putnam 2007 A5: Finite group $n$ elements order $p$, prove either $n=0$ or $p$ divides $n+1$

Putnam 2007 Question A5: "Suppose that a finite group has exactly $n$ elements of order $p$, where $p$ is a prime. Prove that either $n=0$ or $p$ divides $n+1$." I split this problem into two cases: ...
0
votes
2answers
49 views

If $P,Q$ are $p-$subgroup of $G$, then so is $PQ$. [closed]

Let $G$ be a finite group and $p$ divide $|G|$. Suppose $P$ and $Q$ are two subgroups s.t. $P\subset N_G(Q)$. Prove that $PQ$ is a $p-$subgroup of $G$. I imagine that I have to use second isomorphism ...
0
votes
0answers
29 views

Coloring the vertices of a decagon

In how many ways can you color, up to symmetry, the vertices of a regular decagon using $q$ colors? (We are talking here about the dihedral group of order $20$). So I was thinking about maybe group ...
1
vote
1answer
70 views

Let $G=\Bbb Z_{30} \times \mathbb Z_{36}$.

Let $G=\Bbb Z_{30}\times \mathbb Z_{36}$. Divide G to direct product of cyclic gropus from order exponent of prime. Than find all the fairs of generators of G. I thought about $36\times30=1080=2^3\...
3
votes
1answer
63 views

Let G be $\mathbb Z_p\times\dots\times \mathbb Z_p$ . Find A(G).

Let G be $\underbrace{\mathbb Z_p\times\dots\times \mathbb Z_p}_{n \text{ times}}$. Find $A(G)$. I know that $A(G)\cong GL_n(\mathbb Z_p)$. I prove it by taking $\varphi$ from $A(G)$ and show that ...
5
votes
1answer
66 views

Flaw in my classification of groups of order 2015

In an attempt to classify groups of order $2015 = 5 \cdot 13 \cdot 31$, I deduced that only $\mathbb{Z}/2015 \mathbb{Z}$ was the only such group. I then checked with some sources that informed me that ...
4
votes
2answers
69 views

How many group homomorphisms we can get from ${\mathbb Z}_{20}$ to ${\mathbb Z}_{10}$?

How many group homomorphisms we can get from ${\mathbb Z}_{20}$ to $ {\mathbb Z}_{10}$? My Try: I think $4$. Because $1 , 2 , 5 , 10$ are the only possible order of image of ${\mathbb Z}_{20} $. ...
3
votes
2answers
61 views

Finite $p$-group with cyclic abelianization is abelian

Is the following result true? If so, how to prove it? Let $G$ be a finite $p$-group whose abelianization $G^{\text{ab}} = G/[G,G]$ is cyclic. Then, $G$ is abelian. I found some similar results ...