The study of symmetry: groups, subgroups, homomorphisms, group actions.
103
votes
18answers
8k 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 ...
80
votes
2answers
2k 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 $G\rightarrow H$?
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? ...
48
votes
4answers
1k 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 ...
46
votes
1answer
1k 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 ...
45
votes
0answers
878 views
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
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 for groups for which this ...
40
votes
5answers
1k 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 ...
39
votes
8answers
736 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 ...
38
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 ...
35
votes
1answer
887 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 ...
34
votes
4answers
856 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 ...
33
votes
1answer
1k 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 ...
32
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
4answers
1k 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 ...
31
votes
3answers
1k 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 ...
29
votes
5answers
1k 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 ...
29
votes
2answers
378 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:
...
28
votes
3answers
803 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 ...
27
votes
0answers
1k 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: ...
25
votes
3answers
319 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 ...
25
votes
2answers
565 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 ...
25
votes
2answers
404 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 ...
25
votes
0answers
494 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 ...
24
votes
8answers
2k 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" ...
24
votes
3answers
502 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 ...
23
votes
3answers
575 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 ...
23
votes
1answer
529 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 ...
23
votes
3answers
622 views
Galois groups of polynomials and explicit equations for the roots
Lets say I have calculated the galois group of some polynomial and I also have the subgroup structure. What's an effective procedure to turn the group into equations for the actual roots of the ...
22
votes
5answers
887 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?
22
votes
4answers
1k 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 ...
22
votes
1answer
433 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 ...
21
votes
6answers
507 views
How do you prove that a group specified by a presentation is infinite?
The group:
$$
G = \left\langle x, y \; \left| \; x^2 = y^3 = (xy)^7 = 1\right. \right\rangle
$$
is infinite, or so I've been told. How would I go about proving this? (To prove finiteness of a ...
21
votes
4answers
520 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 ...
21
votes
2answers
360 views
Can $G≅H$ and $G≇H$ in two different views?
Can $G≅H$ and $G≇H$ in two different views?
We have two isomorphic groups $G$ and $H$, so $G≅H$ as groups and suppose that they act on a same finite set, say $\Omega$. Can we see $G≇H$ as permutation ...
21
votes
1answer
181 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 ...
21
votes
2answers
802 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 ...
21
votes
2answers
383 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
1answer
298 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
3answers
604 views
In a group we have $abc=cba$. Is it abelian?
Let $G$ be a group such that for any $a,b,c\ne1$:
$$abc=cba$$
Is $G$ abelian?
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 $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$?
20
votes
2answers
283 views
About translating subsets of $\Bbb R^2.$
I'm looking for a pair of sets $A,B$ of points in $\Bbb R^2$ such that
$A$ is a union of translated (only translations are allowed) copies of $B;$
$B$ is a union of translated copies of $A;$
$A$ is ...
19
votes
5answers
802 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 ...
19
votes
3answers
2k 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 ...
19
votes
2answers
488 views
Surprising but simple group theory result on conjugacy classes
I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$.
This seems to me like an astonishing ...
19
votes
1answer
436 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 ...
19
votes
1answer
140 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 ...
18
votes
7answers
2k views
Is addition more fundamental than subtraction?
When I followed an introductory! course group theory and throughout all my Math courses as a physicist, subtraction was always defined in terms of the inverse element and addition.
Is this the only ...
18
votes
2answers
962 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 ...
18
votes
2answers
428 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 ...
17
votes
4answers
987 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?
