The study of symmetry: groups, subgroups, homomorphisms, group actions.
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 ...
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 ...
18
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
5answers
798 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 ...
5
votes
3answers
268 views
Prove that if $g^2=e$ for all g in G then G is Abelian.
This question is from group theory in Abstract Algebra and no matter how many times my lecturer teaches it for some reason I can't seem to crack it.
(please note that $e$ in the question is the ...
11
votes
7answers
957 views
Why $PSL_3(\mathbb F_2)\cong PSL_2(\mathbb F_7)$?
Why are groups $PSL_3(\mathbb{F}_2)$ and $PSL_2(\mathbb{F}_7)$ isomorphic?
Update. There is a group-theoretic proof (see answer). But is there any geometric proof? Or some proof using octonions, ...
14
votes
5answers
991 views
Examples and further results about the order of the product of two elements in a group
Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$?
Wikipedia says "not much":
There is no general formula relating the order of a ...
6
votes
2answers
1k views
$|G|>2$ implies $G$ has non trivial automorphism
Well, this is an exercise problem from Herstein which sounds difficult:
How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism?
The only thing, i know which connects a group ...
8
votes
2answers
1k views
Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
I'm having difficulty with exercise 1.43 of Lang's Algebra. The question states
Let $H$ be a subgroup of a finite abelian group $G$. Show that $G$ has a subgroup that is isomorphic to $G/H$.
...
6
votes
3answers
465 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 ...
9
votes
5answers
2k views
Order of elements in abelian groups
How can I prove that if G is an abelian group with elements a and b with orders m and n, respectively, then G contains an element whose order is the least common multiple of m and n?
It's an exercise ...
5
votes
1answer
138 views
Isomorphism between $I_G/I_G^2$ and $G/G'$
Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it.
$G$ is ...
1
vote
1answer
317 views
Irreducible representations of a cyclic group over a field of prime order
Consider $G$ a cyclic group of order $n$ with prime $p\not|n$.
How do I construct all the irreducible representations over $\mathbb F_p$?
How many irreducible representations are there and what are ...
3
votes
2answers
1k views
Prove that the center of a group is a normal subgroup
Let $G$ be a group. We define $H$ where $H$ is the center of/centralizer of $G$:
$$H=\{h\in G| \forall g\in G: hg=gh\}.$$
Prove that $H$ is a (normal) subgroup of $G$.
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 ...
17
votes
7answers
4k views
How to show every subgroup of a cyclic group is cyclic?
I'm teaching a group theory course now, and I wanted to give my students a proof that every subgroup of a cyclic group is cyclic. The easiest way I could think to do this is to say that any cyclic ...
16
votes
4answers
2k 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 ...
3
votes
12answers
2k views
13
votes
4answers
793 views
Alternative proof that the parity of permutation is well defined?
I learned the following theorem about the properties of permutation from Gallian's Contemporary Abstract Algebra.
When I tried to reconstruct the proof myself, I found that it suffices to prove the ...
12
votes
3answers
2k views
Group where every element is order 2
Let $G$ be a group where every non-identity element has order 2.
If |G| is finite then $G$ is isomorphic to the direct product $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \ldots \times ...
5
votes
1answer
227 views
The set of all $x$ such that $xHx^{-1}\subseteq H$ is a subgroup, when $H\leq G$
I found this problem in a textbook of abstract algebra:
Let $H$ be a subgroup of $G$. Prove that $$\{x\in G:xax^{-1}\in H\text{ for every }a\in H\}$$ is a subgroup of $G$.
It's easy to prove ...
5
votes
3answers
385 views
Showing non-cyclic group with $p^2$ elements is Abelian
I have a non-cyclic group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, ...
2
votes
2answers
260 views
Let $G$ be a finite group and $H\triangleleft G$ a normal subgroup. Prove that $|G/H| =|G|$ if, and only if, $H = \{e\}$.
The group $G$ is a finite group, a group with finite number of elements, and $H\triangleleft G $a normal subgroup. How can we prove that the index $|G/H|=|G|$ iff $H=\{e\}$, the identity element?
1
vote
5answers
240 views
Whether $P\cap Q$ and $P\cup Q$ are subgroups
$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup. Is $P \cup Q$ a subgroup?
12
votes
2answers
1k views
Yoneda-Lemma as generalization of Cayley`s theorem?
I came across the statement, that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations.
How exactly does generalizes ...
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 ...
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 ...
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 ...
7
votes
2answers
601 views
Is there a systematic way of finding the conjugacy class and/or centralizer of an element?
Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with "special groups" such as $S_n$ or $A_n$? Are there any ...
11
votes
4answers
1k views
Derived Subgroups and Commutators
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 ...
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
677 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
3answers
539 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 ...
6
votes
2answers
262 views
Is there a free subgroup of rank 3 in $SO_3$?
There are known free subgroups of rank 2 in the set of rotations about the origin in $\mathbb{R}^3$, $SO_3$. For instance, the rotations by angle $\arccos \frac {1}{3}$ about the $z$- and $x$-axis ...
5
votes
2answers
489 views
A Nontrivial Subgroup of a Solvable Group
Question: Let $G$ be a solvable group, and let $H$ be a nontrivial normal subgroup of $G$. Prove that there exists a nontrivial subgroup $A$ of $H$ that is Abelian and normal in $G$.
[ref: this is ...
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.
6
votes
2answers
245 views
How can I prove that, in this group, elements of this form are distinct?
In the group $G=\langle x,y,z\mid xy=yx,zx=x^2z,zy=yz\rangle$, how can I prove that the elements of the form $x^iy^jz^k$ are all distinct?
As Arturo is asking I'm editing this question: I want to ...
17
votes
1answer
639 views
Rubik's Cube Not a Group?
I read online that
although the 3x3x3 is a great example of a mathematical group, larger cubes aren't groups at all.
How can that be true? There is obviously an identity and it is closed, so ...
9
votes
1answer
274 views
If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian?
I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample.
2
votes
2answers
216 views
free groups: $F_X\cong F_Y\Rightarrow|X|=|Y|$
I'm reading Grillet's Abstract Algebra. Let $F_X$ denote the free group on the set $X$. I noticed on wiki the claim $$F_X\cong\!\!F_Y\Leftrightarrow|X|=|Y|.$$ How can I prove the right implication ...
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
433 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 $|\operatorname{Aut}{(H)}|>|\operatorname{Aut}{(G)}|$?
15
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 ...
11
votes
1answer
326 views
When can a (finite) group be written as the quotient of some other group by its center?
So if I'm given $H$, when I can conclude there is a group $G$ such that $H\cong G/Z(G)$? It's easy to show that non-trivial cyclic groups are not of this form. More generally, any group with the ...
10
votes
2answers
438 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 ...
10
votes
3answers
602 views
Coloring the faces of a hypercube
I will restate the 3-D version of the problem. In how many ways can you color a regular cube with 2 colors up to a rotational isometry. The answer is of course a special case of Burnsides Lemma which ...
23
votes
3answers
621 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 ...
