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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1answer
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Number of congruence relations of a 4-element non-cyclic group

How many congruence relations does a 4-element non-cyclic group have? Am I right that I have to find the normal subgroups in order to find the congruence relations? Thanks
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32 views

Question about the equivalence of two linear representations.

I would like to know if this approach is correct. I have two distinct permutation representations and I have to prove that the associated linear representations are equivalent. In order to do this I ...
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1answer
107 views

Probability that $xy = yx$ in a random finite group

let $G$ a finite group, not abelian. I don't know if a short proof of this fact exists : $$\mathbb{P}(xy = yx) \leq 5/8$$ $x,y$ are randomly picked. Edit : If possible, i want to know if there is a ...
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0answers
23 views

Inverse of zero missing for all finite fields F2

I am having a little touble with finite fields at the moment. I am just working from a high school text wich says that the inverse of an element in a group is unique, which to me implies that all ...
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4answers
45 views

Concerning Cyclic Groups

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution). Here it is: ...
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1answer
54 views

$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$

I 've just read from An introduction to the theory of groups from Rotman's the following theorem: Theorem 2.19. Let $p$ be a prime. A group $G$ of order $p^n$ is cyclic if and only if it is an ...
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1answer
36 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
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1answer
26 views

Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
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2answers
28 views

To prove $H:=\{\sigma\in S_n:\sigma(n)=n\} \cong S_{n-1} $ [closed]

Let $H:=\{\sigma\in S_n:\sigma(n)=n\}$ , then $H$ is obviously a subgroup of $S_n$ . I can intuitively feel that $H$ is isomorphic to $S_{n-1}$ but how can I prove it rigorously , Please help .
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1answer
31 views

SL(2,5) and SL(2,11)

there is a problem in my textbook as follows: Why the finite group $SL(2,5)$ is isomorphic to a subgroup of $SL(2,11)$? Thanks for the answers
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1answer
32 views

Group actions and permutation representation

Im trying to solve this problem from Dummit & Foote: Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma\in G$ ...
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1answer
23 views

Question about sum of abelian groups.

So there's a statement in Lang that I would like to understand better. It's contained in his proof of the following statement: Every finite abelian p-group is isomorphic to a product of cyclic ...
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59 views

Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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1answer
30 views

Equality of subgroups of finite cyclic groups

My abstract algebra text has the following proof of the statement, "Let $G$ be a cyclic group with $n$ elements and generated by $a.$ Then $\langle a^s \rangle = \langle a^t \rangle \iff \gcd(s,n) = ...
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0answers
49 views

Supersolvable groups of order $pq^m$

According to Burnside's classification in his book "Theory of groups of finite order", one of the types of non-abelian groups of order $pq^2$ ($p$ and $q$ are distinct primes), has the presentation ...
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2answers
45 views

Sum of the elements of a cyclic subgroup

Let $G$ be a finite cyclic subgroup of the group of units of a commutative unital ring $R$. What is the sum of the elements of $G$, i.e. $$\sum_{x \in G}{x}?$$ [The answer is not difficult in the ...
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45 views

Group theory question (on Nilpotent Groups)

use this notation for the following $\textbf{Theorem}$ - $\textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$ $\textbf{Theorem}$- In a finitely generated ...
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1answer
172 views

To show from definitions , if $|G|=15$ then $G$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$

How to show that any group of order $15$ is isomorphic to $\mathbb Z_3 \times \mathbb Z_5$ ? Please don't use results like "every group of order $15$ is abelian , cyclic " etc. just the definitions . ...
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1answer
31 views

Alternative to the Frattini argument

If $G$ is a finite group with $H \trianglelefteq G$, and $P$ is a Sylow $p$-subgroup of $H$, then we can show that $G = N_G(P)H$. While I'm now aware of the (admittedly much simpler/nicer) Frattini ...
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46 views

any group of order $15$ has an element of order $5$ , without Cauchy's theorem [duplicate]

Without using Cauchy's theorem , can we tell that any group of order $15$ has an element of order $5$ ?
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2answers
106 views

What is known about automorphism group cardinality?

What is known about automorphism group in general and about $|\text{Aut}(G)|$? Is it true that $|\text{Aut}(G)| \le |G|$? Exist any algorithm to build $\text{Aut}(G)$ for given $G$? $G$ is finite.
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1answer
39 views

Generators of $PSL(3,2)$

Is it true that a set of generators for $PSL(3,2)\simeq SL(3,2)$ is: $$\alpha=\left(\begin{array}{ccccccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array}\right),$$ ...
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0answers
48 views

A doubt in M. Hall paper(On the number of Sylow subgroups in a finite group). Please help.

It is the equation 2.5 in the theorem 1 of the paper Hall. I am mentioning the theorem below- Theorem ([M. Hall]) Let $K \unlhd G$, $P \in Syl_p(G)$, then $n_p=a_pb_pc_p$, where $a_p = ...
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1answer
38 views

Order of Group with Elements of Order 2 [duplicate]

Let G be a finite group such that every element in G which isn't the identity has order of 2. Show that $|G| = 2^{n}$ for some $n \in \mathbb{N}$. I know that G is necessarily going to be abelian. ...
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A question on $p$-groups, and order of its commutator subgroup.

$\textbf{QUESTION-}$ Let $P$ be a p-group with $|P:Z(P)|\leq p^n$. Show that $|P'| \leq p^{n(n-1)/2}$. If $P=Z(P)$ it is true. Now let $n > 1$, then If I see $P$ as a nilpotent group and ...
4
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1answer
76 views

how to begin self study of computational group theory.

Today in class on finite group theory our professor taught us Mathieu groups and so we dealt with Steiner system and similar. He said from here on you can pursue computational group theory and start ...
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41 views

Show $G=[A,B]$.

Question- Let $G=AB$ $ $ where $A$ and $B$ are abelian subgroups. Show $G'=[A,B]$. $\textbf{Try}$- As $A$ and $B$ are subgroups then by a lemma in Isaacs (4.1) $[A,B]\ \unlhd\ <A,B>=G$. So ...
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0answers
29 views

Parametrizing a group element

I'm looking at the problem of parametrizing a group element $g \in SL(n,\mathbb{R}).$ I think I understand the concept of parametrization $-$ just a change of coordinates, right? But I don't ...
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3answers
134 views

How to find [G:H]?

Let $F$$=GF(11)$ be finite field of 11 elements. G is group of all non-singular n$\times$n matrices over F.$H$ is subgroup of those matrices whose determinant is 1. Then $[G:H]$=?
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3answers
80 views

Cardinal of a group $G$ such that for all $x\in G$ we have $x^2=e$

Let $G$ be a group such that for all $x\in G$ we have $x^2=e$. Show that if $G$ is finite then the order of $G$ is $2^n$. Here is the solution I have seen in a book. If G is finite, it can be ...
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1answer
34 views

The number, up to isomorphism, or abelian grips of order 40 is

The number, up to isomorphism, or abelian groups of order 40 is: I got: 2*2*10 2*20 40 So the total number is 3. However, the answer says 7, where 40 10*4 8*5 20*2 10*2*2 5*4*2 I think the ...
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1answer
36 views

If $G$ is a group of order $48$, show that the intersection of any two distinct Sylow $2$-subgroups has order $8$

All I know is that we have $3$ Sylow-$2$ subgroups of order $16$. $$o(H \cap K)= o(H)o(K)/o(HK)$$ How to proceed further?
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1answer
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Find\construct a group of order $q(q-1)$ s.t. …

Problem- Let $q$ be a power of a prime $p$ say $q=p^k$. Show that there exists a group $G$ of order $q(q-1)$ with a normal elementary abelian subgroup of order $q$ and such that all elements of order ...
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2answers
58 views

Show that every proper subgroup of this group is finite.

Let $G$ be the group of rational numbers in $[0,1)$ whose denominator is a power of $2$: \begin{align*} G &= \{r/2^k : \text{$r \in \mathbb Z$, $0 \le r < 2^k$, $k = 0, 1, ...
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42 views

last part of proof of schur zassenhaus theorem.

Theorem states- Let $G$ be a finite group of order $mn$ and $N$ be a normal subgroup of order $n$, then schur zassenhaus states that there exist a complement of $N$ in $G$ of order $m$ and all such ...
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1answer
37 views

Consequence of Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$

Lemma: If G is abelian with exponent n, then $|G|\big\vert n^m$ for some $m\in N$. Theorem to be proved: Suppose G is finite abelian and group of order m, let p be a prime number dividing m. Then G ...
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1answer
25 views

How to find orbits and isoropy group?

About this problem ${a}$, I am wondering if there are 5 orbits in $A$? The 5 orbits separately contain elements which 3 are all the same, 2 of 3 are the same and all 3 are different? I am confused ...
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3answers
113 views

Prove that $n_p(N)$ divides $n_p(G)$

Let $N$ be a normal subgroup of $G$ where $G$ is finite group, then we have to prove $n_p(N)$ divides $n_p(G)$ ( here $n_p(G)$ means number of sylow $p$-subgroups of $G$) I was able to prove that ...
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1answer
28 views

Group of order $|G|=pqr$, $p,q,r$ primes has a normal subgroup of order

Let $p,q,r$ be positive primes, $p<q<r$, and let $G$ be a group with $|G|=pqr$. Show that there exists a normal subgroup $H$ of $G$ of order $qr$. I've seen this post Groups of order $pqr$ and ...
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0answers
33 views

Give a $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$

Consider $SL_{2}(\Bbb Z_p)$ if q & p be two primes, $p>q$. Give an example of a subgroup $H\le SL_{2}(\Bbb Z_p)$ such that $|H|=q$ when i) $q|(p-1)$ ii) $q|(p+1)$
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38 views

Conjugacy classes in non-solvable group

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. Also suppose $G$ non-solvable group, $N\unlhd G$, $G/N$ is abelian, $|G/N|=6$ and ...
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1answer
39 views

Elementary abelian $p$-group

How can we show that if $N$ is abelian and $C_G(N)=N$, then $N$ is an elementary abelian $p$-group for some prime $p$?
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1answer
35 views

Classification of groups of order $p^2q$

I have done the classification of groups of order $p^2q$, where $p>q$. $p,q$ odd distinct prime If $P\in Syl_{p}(G)$ & $Q\in Syl_{q}(G)$ then Case $1$: $P=\Bbb Z_{p^2}$. I have done. But my ...
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0answers
46 views

Conjugacy classes in non-abelian simple group

Can we say that every non-abelian simple group has at least 4 non- identity classes?
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1answer
48 views

Question about the answer to Kac's problem: 'Can one hear the shape of a drum?'.

I'm looking at the article of Gordon, Webb and Wolpert http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, having only basic notions of group theory. In this article the authors describe ...
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39 views

Subgroups and justification

Which of these subsets are subgroups of the given group and justify your answer. The group $R^+$ of postive reals under multiplication. The subset $H=(3n|$ $n\in Z^+)$. The group of nonzero ...
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2answers
95 views

Are all torsion groups finite groups?

Are all torsion groups finite groups? I've been trying to find a counter example, but have had no luck so far. Can anyone throw me one, or give me an idea to prove this?
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1answer
39 views

Groups of Order 2 with subgroups

Let G be an abelian group and $a,b\in G$ be two distinct elements with a and b or order $2$. Show that $H=\{e,a,b,ab\}$ forms a subgroup and write out its multiplication table. Justify why all the ...
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1answer
23 views

finite semigroup on one generator,cycle, tail,group,zero element

Suppose we have a finite semigroup on one generator. It has a tail of length r and cycle of length c.The cycle is a group, but what can be chosen as a neutral element of it?Why is not ANY element ...
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1answer
48 views

Solvability of a group

What is the intuition behind the solvable groups? It is defined by composition series. Is there any intuitive way to understand it?