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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137
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2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
132
votes
1answer
4k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
81
votes
1answer
2k views

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which ...
63
votes
2answers
5k 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? ...
46
votes
8answers
4k 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 ...
46
votes
3answers
922 views

How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation ...
34
votes
3answers
508 views

If $\lvert\operatorname{Hom}(H,G_1)\rvert = \lvert\operatorname{Hom}(H,G_2)\rvert$ for any $H$ then $G_1 \cong G_2$

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, $\lvert\operatorname{Hom}(H,G_1)\rvert = \lvert\operatorname{Hom}(H,G_2)\rvert$. How can I show that $G_1 \cong G_2$ ?
31
votes
2answers
461 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
30
votes
3answers
604 views

Alternative proofs that $A_5$ is simple

What different ways are there to prove that the group $A_5$ is simple? I've collected these so far: By directly working with the cycles: page 483 of ...
29
votes
9answers
5k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
29
votes
1answer
370 views

Sudokus as composition tables of finite groups

If $G$ is a finite group then the composition table of $G$ is a latin square (ie, each row and column contains each group element exactly once). Assume now that $|G| = n^2$ for some natural number ...
26
votes
1answer
1k views

Six Frogs - Puzzle

I had come across a puzzle: The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its ...
25
votes
6answers
924 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 $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$?
25
votes
4answers
3k 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 ...
25
votes
1answer
249 views

Smallest order for finite group that needs many elements to generate it

Let $f(n)$ denote the order of the smallest finite group which cannot be generated with less than $n$ elements. Trivially $f(n) \leq 2^n$ since ${\mathbb F}_2^n$ can be seen as a vector space with ...
25
votes
1answer
347 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 ...
25
votes
1answer
2k views

Isomorphic quotients by isomorphic normal subgroups

In this recent question, Iota asked if, given a finite group $G$ and two isomorphic normal subgroups $H$ and $K$, it would follow that $G/H$ and $G/K$ are isomorphic. This is not true (a simple ...
25
votes
2answers
550 views

What does the group ring $\mathbb{Z}[G]$ of a finite group know about $G$?

The group algebra $k[G]$ of a finite group $G$ over a field $k$ knows little about $G$ most of the time; if $k$ has characteristic prime to $|G|$ and contains every $|G|^{th}$ root of unity, then ...
22
votes
6answers
428 views

Generic elementary group theory problems.

This question is about generic group theory problems. here are examples for what I’m referring to: Prove that any group of order $p^2$, where $p$ is a prime, is abelian. Let $G$ be a ...
22
votes
1answer
644 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 ...
21
votes
2answers
515 views

There are at most two prime numbers dividing $|G|$

Need just hints Let $G$ is a finite non-abelian group such that all its proper subgroups are abelian. Then there are at most two different prime numbers dividing $|G|$. I found some ideas about ...
21
votes
2answers
482 views

Lower bounds on the number of elements in Sylow subgroups

Let $p$ be a prime and $n \geq 1$ some integer. Furthermore, let $G$ be a finite group where $p$-Sylow subgroups have order $p^n$. Denote by $n_p(G)$ the number of Sylow $p$-subgroups of $G$. Denote ...
20
votes
7answers
2k 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 ...
19
votes
1answer
317 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 ...
18
votes
5answers
268 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
18
votes
4answers
1k views

Number of subgroups of prime order

I've been doing some exercises from my introductory algebra text and came across a problem which I reduced to proving that: The number of distinct subgroups of prime order $p$ of a finite group ...
18
votes
2answers
481 views

Groups and generating sets

This question feels completely trivial and I am somewhat embarrassed to be asking it, but I am having a brain dead moment and failing to prove what I'm sure is a completely trivial statement about ...
18
votes
1answer
414 views

Is there a geometric idea behind Sylow's theorems?

I have a confession to make: none of the proofs of Sylow's theorems I saw clicked with me. My first abstract algebra courses were more on the algebraic side (without mention of group actions and ...
17
votes
2answers
360 views

Can we uniquely determine a group given the orders of its elements?

Given a finite group $G$ and its order, consider a scenario in which we also know the orders of each of its elements. Does this information alone uniquely determine the group? If not, can we at ...
17
votes
4answers
3k views

How does one compute the sign of a permutation?

The sign of a permutation $\sigma\in \mathfrak{S}_n$, written ${\rm sgn}(\sigma)$, is defined to be +1 if the permutation is even and -1 if it is odd, and is given by the formula $${\rm sgn}(\sigma) ...
17
votes
3answers
575 views

How “abelian” can a non-abelian group be?

Something I have been wondering: in general, is there a bound for how many elements in a finite non-abelian group $G$ can commute with every other element? Equivalently, is there is a bound for the ...
17
votes
2answers
942 views

exercise in Isaacs's book on Character Theory

I'm stuck on an exercise in Isaacs's book "Character Theory of Finite Groups" - it relates to something I'm looking at as part of ongoing research, but I guess it belongs here rather than on ...
17
votes
1answer
408 views

Probability that two randomly chosen permutations will generate $S_n$.

Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$. Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle ...
17
votes
0answers
353 views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In ...
16
votes
4answers
4k 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?
16
votes
5answers
577 views

Upper bounds on the size of $\operatorname{Aut}(G)$

Any automorphism of a group $G$ is a bijection that fixes the identity, so an easy upper bound for the size of $\operatorname{Aut}(G)$ for a finite group $G$ is given by ...
16
votes
3answers
1k views

Constructions of the smallest nonabelian group of odd order

I write $|X|$ for the number of elements in a finite set $X$. Recall some basic facts: If $p$ is a prime number, then any group $G$ of order $p^2$ is abelian. Sketch of proof: Fix a prime $p$ ...
16
votes
1answer
392 views

Books to understand the construction of all groups of a specific order

The algorithms introduced by Besche–Eick (1999) were used to construct (or count) the groups of order up to 2000 in Besche–Eick–O'Brien (2002), yet I find the algorithms somewhat inaccessible. How ...
16
votes
1answer
422 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 $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$. ...
16
votes
3answers
575 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 ...
16
votes
2answers
387 views

Writing a group element as $ghg^{-1} h^{-1}$ and as $g^2 h^2$

I recently read the elegant paper Generalized Frobenius Schur Numbers, by Bump and Ginzburg, which I learned about here. The results in this paper imply the following: Let $G$ be a finite group ...
15
votes
1answer
708 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. ...
15
votes
1answer
444 views

When is $\mathfrak{S}_n \times \mathfrak{S}_m$ a subgroup of $\mathfrak{S}_p$?

Inspired by another question, I wondered when $\mathfrak{S}_n \times \mathfrak{S}_m$ is isomorphic to a subgroup of $\mathfrak{S}_p$. Eliminating the obvious cases, the question becomes: Let ...
15
votes
2answers
325 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 ...
15
votes
2answers
260 views

How many different elements can we obtain by multiplying all element in a group?

Let $G$ be a finite group. How many different elements can we obtain by multiplying all element in a group? Of course, if $G$ is abelian the answer is one but when G is non-abelian, changing the ...
15
votes
1answer
397 views

What are the symmetries of a colored rubiks cube?

Technically the symmetry group of the rubiks cube is the symmetry group of the cube with all its label peeled off. The normal rubiks cube with all its faces painted different colors has trivial ...
14
votes
1answer
326 views

Does $abab=baba$ imply commutativity in a Group of uneven order?

Suppose $(G,\cdot)$ is a finite group of uneven order such that $abab=baba$ for any $a,b\in G$. Does this mean that $G$ is commutative?
14
votes
4answers
2k 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 ...
14
votes
2answers
1k views

Isomorphic quotient groups

Suppose that G is a finite group, and that H and K are normal subgroups of G with trivial intersection, and suppose that H and K are isomorphic. Is it true that the quotient groups G/H and G/K are ...
14
votes
5answers
3k 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. ...