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
0answers
3 views

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 ...
0
votes
1answer
24 views

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.
0
votes
2answers
63 views

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 ...
1
vote
1answer
14 views

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 ...
2
votes
2answers
31 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 $D_{n!/2}$?
3
votes
1answer
37 views

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 $G$, ...
3
votes
0answers
46 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 ...
3
votes
1answer
47 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? ...
0
votes
5answers
70 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, ...
0
votes
4answers
50 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 ...
1
vote
1answer
35 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
0
votes
0answers
23 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 ...
1
vote
0answers
20 views

Can we find a one-dimensional groupe which is isomorphic to $C(ℚ)^\mathrm{tors}$?

Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the ...
6
votes
1answer
89 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 ...
1
vote
0answers
22 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 ...
2
votes
4answers
41 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: ...
5
votes
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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
19 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 ...
0
votes
2answers
27 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 .
1
vote
1answer
29 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
1
vote
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 $\sigma\in G$ ...
1
vote
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 ...
2
votes
0answers
57 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 ...
0
votes
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) = ...
0
votes
2answers
39 views

Number of Homomorphisms [on hold]

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 ...
1
vote
0answers
41 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 ...
0
votes
2answers
44 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 ...
1
vote
0answers
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 ...
1
vote
1answer
120 views
+50

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 . ...
1
vote
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 ...
0
votes
0answers
45 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 $5$ ?
3
votes
2answers
97 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.
1
vote
1answer
33 views

Generators of $PSL(3,2)$

Is it true that a set of generators for $PSL(3,2)\simeq SL(3,2)$ is: $$\alpha=\left(\begin{array}{ccccccc} 0&0&1\\ 0&1&0\\ 1&0&0\\ \end{array}\right),$$ ...
0
votes
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 = ...
0
votes
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. ...
4
votes
0answers
54 views
+50

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
votes
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 ...
0
votes
0answers
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 ...
2
votes
0answers
27 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 ...
0
votes
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 $[G:H]$=?
5
votes
3answers
75 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 ...
0
votes
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 ...
1
vote
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?
4
votes
1answer
40 views

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 ...
3
votes
2answers
56 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, ...
-4
votes
0answers
41 views

To prove any two non-abelian groups of order $pq$ are isomorphic , without Sylow theorems [closed]

How to prove , without using Sylow theorems , that any two non-abelian groups of order $pq$ , where $p,q$ are distinct primes , are isomorphic ?
0
votes
0answers
39 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 ...
0
votes
0answers
29 views

Centralizer of unique cyclic subgroup of order equal to exponent of group [closed]

Let $G$ a finite group and $K\leq G$ the unique cyclic subgroup of $G$ with $|N|=\exp(G)$. Is $C_{G}(N)=N$?
1
vote
1answer
35 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 ...