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.
11
votes
4answers
2k 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?
8
votes
1answer
400 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
3answers
461 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 $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$
...
16
votes
7answers
1k 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 ...
2
votes
6answers
814 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.
...
2
votes
1answer
141 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."
...
17
votes
1answer
257 views
Finite Groups with a subgroup of every possible index
Suppose $G$ is a finite group, with $|G|=n$. Suppose also that for every positive integer $m\mid n$, $G$ has a subgroup of index $m$. Are there any general statements (structural or otherwise) I can ...
8
votes
4answers
275 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 ...
11
votes
4answers
676 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 ...
7
votes
2answers
135 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 ...
7
votes
3answers
537 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 ...
4
votes
2answers
1k 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.
1
vote
2answers
185 views
Groups/Linear maps
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 ...
48
votes
2answers
4k 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? ...
19
votes
1answer
432 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 ...
20
votes
6answers
805 views
Existence of a normal subgroup with $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$
Does there exists a finite group $G$ and a normal subgroup $H$ of $G$ such that $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$?
14
votes
4answers
1k 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 ...
10
votes
2answers
434 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 ...
3
votes
3answers
773 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
2
votes
2answers
118 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 $G\cong H$?
5
votes
5answers
201 views
Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.
Let $G$ a group of order $6$. Prove that:
i) $G$ contains 1 or 3 elements of order 2.
ii) $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$.
I haven´t covered Sylow groups and normal groups. ...
3
votes
1answer
103 views
Metric on a group
Is there a non-abelian finite group $G$ with the property: If metric $d$ on $G$ is left invariant then is also right invariant?
3
votes
4answers
306 views
Quaternion group as an extension
I'm trying to understand how the quaternion group Q arises as an extension of $\mathbb{Z}_{4}$ by $\mathbb{Z}_{2}$.
More precisely, I'm trying to find the two homomorphisms in the short exact sequence ...
2
votes
2answers
116 views
$T(G)$ may not be a subgroup?
It is obvious that for an abelian group $G$; the set of all torsion elements: $$T(G)=\{x\in G|x^n=0 \text{, for some nonzero integer } n\}$$ is a subgroup of the group. I am asked to probe this fact ...
1
vote
1answer
142 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. ...
0
votes
4answers
644 views
Any subgroup of index $p$ in a $p$-group is normal.
Let $p$ be a prime number and $G$ a finite group where $|G|=p^n$, $n \in \mathbb{Z_+}$. Show that any subgroup of index $p$ in it is normal in $G$. Conclude that any group of order $p^2$ have a normal ...
14
votes
1answer
301 views
What are all conditions on a finite sequence $x_1,x_2,…,x_m$ such that it is the sequence of orders of elements of a group?
My Definition: The finite sequence $x_1,x_2,...,x_m$ of nonnegative integers, is said to be generated by the finite group $G$ iff
$n:=|G|=x_1+x_2+...+x_m$.
$n$ has $m$ divisors.
if ...
9
votes
3answers
605 views
Generalization of index 2 subgroups are normal
Let $G$ be a finite group and $H$ a subgroup of index $p$, where $p$ is a prime. If $\operatorname{gcd}(|H|, p-1)=1$, then $H$ must be normal. Does somebody have a quick proof of this?
5
votes
2answers
116 views
Estimates on conjugacy classes of a finite group.
In Character Theory Of Finite Groups by I Martin Issacs as exercise 2.18, on page 32.
Theorem:
Let $A$ be a normal subgroup of $G$ such that $A$ is the centralizer of every non-trivial element ...
10
votes
2answers
300 views
Is there an infinite simple group with no element of order $2$?
According to the Feit-Thompson theorem, every group of odd order is solvable and thus every finite nonabelian simple group has even order. Thus every finite nonabelian simple group has an involution ...
7
votes
3answers
374 views
Finite abelian $p$-group with only one subgroup size $p$ is cyclic
My goal is to prove this:
If $G$ is a finite abelian $p$-group with a unique subgroup of size $p$,
then $G$ is cyclic.
I tried to prove this by induction on $n$, where $|G| = p^n$ but was not ...
7
votes
1answer
156 views
Generators for $S_n$
This problem is from Dixon's book telling that; if $x$ is any nontrivial element of the symmetric group $S_n$ and $n≠4$, then there exists an element $y\in S_n $ such that $S_n=\langle x,y\rangle$. ...
7
votes
1answer
612 views
Enumerating all subgroups of the symmetric group
Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group ...
6
votes
2answers
315 views
Classify groups of order 27
Let $|G|=27$. Prove that all subgroups of index 3 are normal. Classify all groups of order 27.
I can do the first one, but the classification is overwhelming. I don't even know where to start.
...
5
votes
2answers
212 views
What is an irrreducible character of a finite group?
Let $S_n$ be the group of permutations of $\{1, 2, \ldots, n\}$. A “character” for $S_n$ is a function $\chi\colon S_n \to \mathbb{C} \setminus \{0\}$ with $\chi(ab) = \chi(a)\chi(b)$ for all $a, b ...
4
votes
3answers
470 views
Question about normal subgroup and relatively prime index
Suppose $G$ is finite, $K$ is a normal subgroup in $G$, $H$ is a subgroup of $G$, and $|K|$ is relatively prime to $[G:H]$. Show that $K$ is a subgroup of $H$.
I don't know where to begin...
4
votes
1answer
127 views
Valuations on number fields
I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, ...
10
votes
1answer
247 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 $2n$ (i.e., the cardinality of the set $G$ is $2n$), show that the number of elements of $G$ of order $2$ is odd.
...
9
votes
2answers
100 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 ...
3
votes
1answer
1k views
No group of order 36 is simple
Fraleigh(7ed) Example37.14 No group of order 36 is simple. Such a group $G$ has either $1$ or $4$ subgroups of order $9$. If there is only one such subgroup, it is normal in $G$. If there are four ...
2
votes
5answers
197 views
If $H\unlhd G$ with $(|H|,[G:H])=1$ then $H$ is the unique such subgroup in $G$.
Here is a problem from "An introduction to the Theory of Groups" by J.J.Rotman:
Let $G$ be a finite group, and let $H$ be a normal subgroup with $(|H|,[G:H])=1$. Prove that $H$ is the unique such ...
2
votes
1answer
1k views
Computing the order of elements in Dihedral Groups
I am working through "Abstract Algebra" by Dummit & Foote. Exercise 1.2.1 states:
Compute the order of each of the elements in ... $D_6$, $D_8$, and $D_{12}$.
I have found that:
...
1
vote
1answer
80 views
Solving conjugacy equations in dihedral groups.
For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that
$a(rR^m ) a^{-1}=R^2$
$b(rR^m ) b^{-1}=r$
$c(rR^m ) c^{-1}=rR$
$D_n$ is dihedral group of an $n$-gon represented by
...
1
vote
1answer
190 views
Normal subgroups of $S_N$
Is there a list of all normal subgroups for $S_N$?
What is a criteria for a finite group to be a normal subgroup of $S_N$?
Which of them are kernels of irreducible representation? From a partition ...
0
votes
1answer
245 views
Prove that every p-group is nilpotent.
Let $G$ be a p-group $|G| = p^n$. Then G is nilpotent in the following sense:
Let $C^1(G) = [G,G] = G'$
And $C^n(G) = [G,C^{n-1}]$.
G is nilpotent iff there is a $n_0$ such that $C^n = \{e\}$.
What ...
-3
votes
2answers
371 views
Non-isomorphic abelian groups of order $19^5$
I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
13
votes
2answers
277 views
Is there a geometric realization of Quaternion group?
Is there a geometric realization of the Quaternion group:
$$Q = \langle i,j,k \mid i^2 = j^2 = k^2 = ijk \rangle$$
I dont think it can be realized as the symmetries/rotations of a 3D shape so could ...
5
votes
1answer
123 views
When does a biregular graph for the free product 2∗(2×2) have a 4 cycle?
I'd like to understand a graph theoretic property in terms of group theory. I have some boring graphs, and some neat graphs, all created from groups, but I don't know how to tell a boring group from ...
4
votes
2answers
101 views
Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$
This question coutesy of Allan Clark's "Elements of Abstract Algebra" (60$\zeta$).
Find the number of homomorphisms from $\Bbb{Z}_m\to \Bbb{Z}_n$ as a function of $m$ and $n$.
This is stumping me, ...
4
votes
8answers
460 views
Order of cyclic groups
Wikipedia says:
It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1.
The statement seems provable ...
