The study of symmetry: groups, subgroups, homomorphisms, group actions.

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158
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21answers
15k views

Nice examples of groups which are not obviously groups

I am searching for some groups, where it is not so obvious that they are groups. In the lectures script there are only examples like $\mathbb{Z}$ under addition and other things like that. I ...
139
votes
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 $G\rightarrow H$?
136
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 } ...
86
votes
2answers
3k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
82
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 ...
66
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? ...
56
votes
4answers
2k views

How is a group made up of simple groups?

I've read more than once the analogy between simple groups and prime numbers, stating that any group is built up from simple groups, like any number is built from prime numbers. I've recently started ...
53
votes
1answer
2k views

How was the Monster's existence originally suspected?

I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence. For ...
51
votes
6answers
2k views

Why are groups more important than semigroups?

This is an open-ended question, as is probably obvious from the title. I understand that it may not be appreciated and I will try not to ask too many such questions. But this one has been bothering me ...
48
votes
12answers
3k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
47
votes
6answers
2k views

What kind of “symmetry” is the symmetric group about?

There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
47
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 ...
47
votes
1answer
2k views

Is Lagrange's theorem the most basic result in finite group theory?

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group ...
46
votes
2answers
1k views

Reference request for tricky problem in elementary group theory

The following could have shown up as an exercise in a basic Abstract Algebra text, and if anyone can give me a reference, I will be most grateful. Consider a set $X$ with an associative law of ...
46
votes
4answers
2k views

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
43
votes
3answers
3k views

The direct sum $\oplus$ versus the cartesian product $\times$

In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times ...
43
votes
1answer
1k views

What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
41
votes
8answers
2k views

Does $G\cong G/H$ imply that $H$ is trivial?

Let $G$ be any group such that $$G\cong G/H$$ where $H$ is a normal subgroup of $G$. If $G$ is finite, then $H$ is the trivial subgroup $\{e\}$. Does the result still hold when $G$ is infinite ? In ...
38
votes
1answer
988 views

Ignoring elements of small order in the simple group of order $60$

The simple group of order $60$ can be generated by the permutations $(1,2)(3,4)$ and $(1,3,5)$, but all you need to do is square the first one and it becomes the identity. Can't we find a version of ...
38
votes
4answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
37
votes
2answers
1k views

Is there a group with exactly 92 elements of order 3?

The number of elements of order 2 in a group is fairly restricted: 0, odd, or infinity. All such possibilities occur already in the trivial group and in dihedral groups. The number of elements of ...
32
votes
1answer
1k views

Automorphisms inducing automorphisms of quotient groups

Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow ...
31
votes
3answers
1k views

Why are there only a finite number of sporadic simple groups?

Is there any overarching reason why, after excluding the infinite classes of finite simple groups (cyclic, alternating, Lie-type), what remains---the sporadic, exceptional finite simple groups, is in ...
31
votes
5answers
2k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
31
votes
2answers
465 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
5answers
2k views

Can every group be represented by a group of matrices?

Can every group be represented by a group of matrices? Or are there any counterexamples? Is it possible to prove this from the group axioms?
30
votes
3answers
638 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
6k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
29
votes
8answers
3k views

Why should we care about groups at all?

Someone asked me today, "Why we should care about groups at all?" I realized that I have absolutely no idea how to respond. One way to treat this might be to reduce "why should we care about groups" ...
29
votes
1answer
382 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 ...
28
votes
4answers
1k views

Does every set have a group structure?

I know that there is no vector space having precisely $6$ elements. Does every set have a group structure?
28
votes
3answers
4k views

Why do we define quotient groups for normal subgroups only?

Let $G \in \mathbf{Grp}$, $H \leq G$, $G/H := \lbrace gH: g \in G \rbrace$. We can then introduce group operation on $G/H$ as $(xH)*(yH) := (xy)H$, so that $G/H$ becomes a quotient group when $H$ is a ...
28
votes
3answers
863 views

Are the axioms for abelian group theory independent?

(I give a lengthy introduction to a concise question -- scroll down if you want to jump straight up to the question). Recall that abelian group theory consists of two primitive symbols: $\cdot$ which ...
28
votes
2answers
313 views

Information-theoretic aspects of mathematical systems?

It occured to me that when you perform division in some algebraic system, such as $\frac a b = c$ in $\mathbb R$, the division itself represents a relation of sorts between $a$ and $b$, and once you ...
27
votes
6answers
562 views

Can $S^2$ be turned into a topological group?

I know that $S^1$ and $S^3$ can be turned into topological groups by considering complex multiplication and quaternion multiplication respectively, but I don't know how to prove or disprove that $S^2$ ...
27
votes
2answers
2k views

Structure Theorem for abelian torsion groups that are not finitely generated

I know about the structure theorem for finitely generated abelian groups. I'm wondering whether there exists a similar structure theorem for abelian groups that are not finitely generated. In ...
27
votes
3answers
868 views

Where does the word “torsion” in algebra come from?

Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry ...
27
votes
1answer
866 views

What can we say of a group all of whose proper subgroups are abelian?

Let $G$ be a group (not necessarily finite). Can we say something about its structure if we suppose that all of its proper subgroups are abelian? Is there a difference between the finite case and the ...
27
votes
2answers
499 views

When is the product of $n$ subgroups a subgroup?

Let $G$ be any group. It's a well-known result that if $H, K$ are subgroups of $G$, then $HK$ is a subgroup itself if and only if $HK = KH$. Now, I've always wondered about a generalization of this ...
26
votes
6answers
2k views

Intuition behind conjugation in group theory

I am learning group theory, and while learning automorphisms, I came across conjugation as an example in many textbooks. Though the definition itself, (and when considering the case of abelian ...
26
votes
4answers
2k views

Center-commutator duality

I'm reading this article by Keith Conrad, on subgroup series. I'm having trouble with a statement he does at page 6: Any subgroup of $G$ which contains $[G,G]$ is normal in $G$. He says this as ...
26
votes
4answers
3k views

Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian

I've been working on this problem listed in Herstein's Topics in Algebra (Chapter 2.3, problem 4): If $G$ is a group such that $(ab)^i = a^ib^i$ for three consecutive integers $i$ for all $a, b\in ...
26
votes
4answers
785 views

Simplicity of $A_n$

I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do ...
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 ...
26
votes
3answers
943 views

What can we learn about a group by studying its monoid of subsets?

If $G$ is a group, then $M(G)=2^G$ is has a monoid structure when we define $AB$ to be $\{ab|a\in A,b\in B\}$ and $1_{M(G)}=\{1\}$. How much of the structure of $G$ can be recovered by studying the ...
25
votes
6answers
934 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
258 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
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 ...
24
votes
6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...