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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Proof that the order of any finite $p$-group is a power of $p$

What is the most concise proof that the order of any finite $p$-group is a power of $p$?
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Cayley's theorem — more than one isomorphism

I've just been learning about Cayley's theorem and a couple of things occurred to me: We know that every finite group of order $n$ is isomorphic to some subgroup of $S_n$. But perhaps there are ...
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25 views

Show that the map $ψ:ℤ^{r}×ℤ/nℤ→ℤ^{r}⊕ℤ/nℤ$ is an isomorphism

Show that the map $$ψ:ℤ^{r}×ℤ/nℤ→ℤ^{r}⊕ℤ/nℤ$$ given by $$(α₁,...,α_{r},R)→ψ(α₁,...,α_{r},R)=∑_{k=1}^{r}α_{k}P_{k}+R$$ is an isomorphism. The same question for $ℤ/2mℤ× ℤ/2ℤ$ whee $m$ is an integer. ...
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2answers
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Can we find a finite abelian group $(A,+)$ such that $A⊂ℝ$

Let us consider the finite abelian group (ℤ/nℤ,+). My question is: Can we find a finite abelian group $(A,+)$ such that $A⊂ℝ$ and $(ℤ/nℤ,+)$ is isomorphic to $(A,+)$ . The same question for the groups ...
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2answers
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Proving that $a$ is a $p$-cycle

I was reading Topic in Algebra by I.N. Herstein and trying to solve a problem from it. If $p$ is a prime number, show that in $S_p$ there are $(p-1)!+1$ elements $x$ satisfying $x^p=e$. I was ...
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On Finite Monoid [duplicate]

Consider a finite monoid $(M,*)$. Let the identity element is the only idempotent element in $M$. Prove that $(M,*)$ is a group.
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Equivalence classes of (2,3)-pairs in PSL(2,q)

Let $G = PSL(2,q)$. I am interested in the different ways of writing $G$ as an image of the modular group. If $A, B \in G$ are elements of order $2, 3$ respectively, then $\langle A,B \rangle$ is an ...
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1answer
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Why $( Z_3\rtimes Z_2)\times Z_2 \cong (Z_3\times Z_2)\rtimes Z_2$?

I got an explanation, it says as $Z_2$ is in the kernel of the homomorphism. But I can't understand from that. Also can you tell me why $Z_3\rtimes Z_2\cong S_3$ ? Thank you.
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2answers
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Does a Group being Finite Imply that It Is Cyclic?

I have been studying Abstract Algebra, and all the finite groups that we have studied so far have also been cyclic. So, is it true that all finite groups are cyclic? If yes, what is the theorem? If ...
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1answer
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Maximal Subgroups in Groups of Order $p^k$

The following question is from a past problem set in a course on group theory. For reference, the text used is by Derek Robinson, entitled "A Course in the Theory of Groups". "Show that in a group ...
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2answers
33 views

Symmetric groups isomorph to dihedral groups.

I've noticed, that $S_2 \cong D_1$ and $S_3 \cong D_3$. Is every symmetric group $S_n$ (no including $S_1$) isomorph to the dihedral group $D_{n!/2}$?
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1answer
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Bounding the order of a group by its nilpotentizer

Let $G$ be a finite non-nilpotent group. We put $nil_G(x)=\{y\in G\mid \langle x,y \rangle \text{ is nilpotent}\}$, called the nilpotentizer of $x$. Note that $nil_G(x)$ may not be a subgroup of $G$, ...
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0answers
49 views

Characters of a finite group

Recently, I have been studying about Character Theory of Finite Groups, mostly from "Groups and Representations" by J. Alperin & R. Bell. In the aforementioned textbook, the characters of a finite ...
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1answer
51 views

Ways to find the order of an element in a group

Is there a better way of finding the order of an element in a group other than circling until the identity is reached? Is there or CAN there be a better general ways of finding orders of elements? ...
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5answers
73 views

Is it true, $O(ab)=O(ba),$ Where $G$ is a group and $a,b \in G.$

Suppose $O(a)$ and $O(b)$ is finite and also $O(ab)$ and $O(ba)$ is finite. Then L.C.M $(|a|,|b|)= L.C.M (|b|,|a|).$ (Is that Correct ?) Suppose $O(a)$ and $O(b)$ is finite but $O(ab)$ is infinite, ...
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4answers
52 views

Show that some group is isomorphic to $\mathbb{Z_n}$

If $G$ has order $4$ and has an element of order $4$, then $G$ is isomorphic to $\mathbb{Z_4}$. Can someone briefly explain why this is true? I understand that $|G| = 4$, but I don't understand ...
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1answer
40 views

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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27 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
97 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
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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
47 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
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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
20 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} $ [on hold]

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
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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
28 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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58 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
28 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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2answers
39 views

Number of Homomorphisms [closed]

I need to find the number of homomorphisms from one set into another and from one set onto another. The resourses that I have looked at are not very clear and the previous questions on this site do ...
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0answers
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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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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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43 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
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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
29 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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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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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
35 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
45 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
35 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 ...
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1answer
69 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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40 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
28 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
133 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
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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
34 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 ...