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

21
votes
6answers
8k views

Normal subgroup of prime index

Generalizing the case $p=2$ we would like to know if the statement below is true. Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal. ...
26
votes
5answers
6k views

Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
20
votes
1answer
2k views

Group of even order contains an element of order 2

I am working on the following problem from group theory: If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd. That is, for some integer $k$, there are ...
12
votes
1answer
2k views

$\operatorname{Aut}(V)$ is isomorphic to $S_3$

I'm currently working my way through Harvard's online abstract algebra lectures (if you're interested, you can find them here). The lectures come complete with notes and homework problems. Of ...
6
votes
2answers
361 views

An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and ...
9
votes
2answers
931 views

Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.

Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic. What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
65
votes
8answers
12k views

Are there real world applications of finite group theory?

I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as ...
14
votes
3answers
2k views

Problem from Herstein on group theory

The problem is: If $ G $ is a finite group with order not divisible by $ 3 $, and $ (ab)^{3} = a^{3} b^{3} $ for all $ a,b \in G $, then show that $ G $ is abelian. I have been trying this for a ...
11
votes
2answers
403 views

Known bounds for the number of groups of a given order.

The number of nonisomorphic groups of order $n$ is usually called $\nu(n)$. I found a very good survey about the values. $\nu(n)$ is completely known absolutely up to $n=2047$, and for many other ...
7
votes
2answers
1k views

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general ...
3
votes
1answer
239 views

If $|A| > \frac{|G|}{2} $ then $AA = G $ [closed]

I'v found this proposition. If $G$ is a finite group , $ A \subset G $ a subset and $|A| > \frac{|G|}{2} $ then $AA = G $. Why this is true ?
13
votes
3answers
530 views

If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.

Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. ...
12
votes
3answers
5k views

A normal subgroup intersects the center of the $p$-group nontrivially

If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
7
votes
2answers
1k views

Product of elements of a finite abelian group

Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$. Prove that if there is more than one element of order $2$ then $x=e$. What I've done so far: (#1 is just for ...
21
votes
2answers
805 views

Which finite groups are the group of units of some ring?

Motivated by this question: Which finite groups are the group of units of some ring? Which finite groups are the group of units of some finite ring? Which finite abelian groups are the group of ...
18
votes
6answers
5k views

Union of the conjugates of a proper subgroup

Let G be a finite group and H be a proper subgroup. Prove that the union of the conjugates of H is not the whole of G. Thanks for any help
16
votes
2answers
865 views

Has this “generalized semidirect product” been studied?

If $G$ is a finite group with subgroups $H$ and $K$ such that $HK = G$ and $H\cap K = \{1\}$ we get that every element of $G$ can be written uniquely as $hk$ with $h\in H$ and $k\in K$. This then ...
25
votes
7answers
4k views

Product of all elements in an odd finite abelian group is 1

This should be an easy exercise: Given a finite odd abelian group $G$, prove that $\prod_{g\in G}g=e$. Indeed, using Lagrange's theorem this is trivial: There is no element of order 2 (since the order ...
12
votes
4answers
765 views

Elementary isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$

At this Wikipedia page it is claimed that to construct an isomorphism between $\operatorname{PSL}(2,5)$ and $A_5$, "one needs to consider" $\operatorname{PSL}(2,5)$ as a Galois group of a Galois cover ...
9
votes
3answers
1k views

Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$

I need help with the following problem: Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
7
votes
3answers
1k views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
6
votes
2answers
1k views

Show that a finite group with certain automorphism is abelian

Let $G$ be a finite group and $f:G\to G$ an isomorphism. If $f$ has no fixed points (i.e., $f(x)=x$ implies $x=e$) and if $f\circ f$ is the identity, then $G$ is abelian. (Hint: Prove that every ...
79
votes
2answers
6k views

More than 99% of groups of order less than 2000 are of order 1024?

In Algebra: Chapter 0, the author made a remark (footnote on page 82), saying that more than 99% of groups of order less than 2000 are of order 1024. Is this for real? How can one deduce this result? ...
33
votes
6answers
5k views

Is there an easy way to see associativity or non-associativity from an operation's table?

Most properties of a single binary operation can be easily read of from the operation's table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a ...
12
votes
4answers
2k views

A kind of converse of Lagrange's Theorem

Let $G$ a finite group. If $a\in G$, a consequence of the Lagrange's Theorem is that the order of $a$ divides the order of $G$. Let $p=|G|$. If $p$ is prime, it is well known that $G$ is cyclic, and ...
10
votes
4answers
4k views

Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ ...
7
votes
2answers
416 views

Does $G\times K\cong H\times K$ imply $G\cong H$?

Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?
17
votes
5answers
3k views

Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
2
votes
1answer
334 views

Definition of $K$-conjugacy classes

I would like to understand the definition of "$K$-conjugacy classes" I found in an article by G. Pazderski Pazderski, Gerhard. "On the number of irreducible representations of a finite group." ...
2
votes
1answer
427 views

Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.

I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. ...
19
votes
4answers
3k views

Derived subgroup where not every element is a commutator

Let $G$ be a group and let $G'$ be the derived subgroup, defined as the subgroup generated by the commutators of $G$. Is there an example of a finite group $G$ where not every element of $G'$ is a ...
18
votes
1answer
520 views

A special subgroup of groups of order $n$

Let $G$ be a group with $|G| = n$ and let $ \emptyset \ne S \subseteq G$. I want to show that $S^n$ is a SUBGROUP of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$. ...
32
votes
1answer
580 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
11
votes
1answer
2k views

Every finite group is isomorphic to some Galois group for some finite normal extension of some field.

Every finite group is isomorphic to some Galois group for some finite normal extension of some field. I'm trying to write a proof, but I know this is incorrect. Can anyone point me in the write ...
15
votes
1answer
3k views

A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$

This is problem 3 from Hungerford's section about the Sylow theorems. I have already read hints saying to use induction and that $p$-groups always have non-trivial centres, but I'm still confused. ...
10
votes
3answers
1k views

Find four groups of order 20 not isomorphic to each other.

Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic. So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral ...
4
votes
2answers
1k views

Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .

I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p). I should note that by simple I mean ...
6
votes
1answer
6k views

Normal subgroups of dihedral groups

In relation to my previous question, I am curious about what exactly are the normal subgroups of a dihedral group $D_n$ of order $2n$. It is easy to see that cyclic subgroups of $D_n$ is normal. But ...
4
votes
1answer
133 views

Generators of $S_n$

Show that $S_n$ is generated by the set $ \{ (12),(123\dots n) \} $. I think I can see why this is true. My general plan is (1) to show that by applying various combinations of these two cycles you ...
3
votes
3answers
324 views

Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
1
vote
1answer
443 views

If $H$ is a subgroup of $G$ of finite index $n$, then under what condition $g^n\in H$ for all $g\in G$

A consequence of Lagrange's theorem in finite group theory is that $g^{|G|}=e$ for all $g\in G$. I wonder whether this be generalized along these lines: If $H$ is a subgroup of $G$ of finite ...
31
votes
4answers
5k views

Finite Groups with exactly $n$ conjugacy classes $(n=2,3,…)$

I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes. If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of ...
23
votes
1answer
784 views

Recovering a finite group's structure from the order of its elements.

Suppose you know the following two things about a group $G$ with $n$ elements: the order of each of the $n$ elements in $G$; $G$ is uniquely determined by the orders in (1). Question: How ...
18
votes
2answers
2k views

Subgroups of a direct product

Until recently, I believed that a subgroup of a direct product was the direct product of subgroups. Obviously, there exists a trivial counterexample to this statement. I have a question regarding ...
8
votes
1answer
3k views

Converse of Lagrange's theorem for abelian groups

I'm trying to prove that the converse of Lagrange's theorem is true for finite abelian groups (i.e. "given an abelian group $G$ of order $m$, for all positive divisors $n$ of $m$, $G$ has a subgroup ...
10
votes
1answer
2k views

If $H$ is a proper subgroup of a $p$-group $G$, then $H$ is proper in $N_G(H)$.

Let $H$ be a proper subgroup of $p$-group $G$. Show that the normalizer of $H$ in $G$, denoted $N_G(H)$, is strictly larger than $H$, and that $H$ is contained in a normal subgroup of index $p$. ...
9
votes
3answers
1k views

Three finite groups with the same numbers of elements of each order

There exist pairs of finite groups $G$ and $H$ such that $G$ and $H$ are not isomorphic, yet they have the same number of elements of each order. For example, if $p$ is an odd prime, then the group ...
8
votes
2answers
363 views

Exponent of $GL(n,q)$.

Another exponent problem. $GL(n,q)$ is the group of invertible $n\times n$ matrices over the finite field $GF(q)$, where $q$ is a prime power. I am trying to figure out the exponent of this group. ...
13
votes
5answers
855 views

Prove that $(a_1a_2\cdots a_n)^{2} = e$ in a finite Abelian group

Let $G$ be a finite abelian group, $G = \{e, a_{1}, a_{2}, ..., a_{n} \}$. Prove that $(a_{1}a_{2}\cdot \cdot \cdot a_{n})^{2} = e$. I've been stuck on this problem for quite some time. Could someone ...
4
votes
1answer
2k views

Dedekind modular law

Dedekind modular law. If $A,B,C$ are subgroups of a group $G$ with $A \subseteq B$ then $A(B \cap C) = B \cap AC$. Below is what I want to prove. Let K be a finite group with $K = LH$, where $L,H$ ...