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

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38
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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 ...
44
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 ...
13
votes
2answers
2k 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 ...
11
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. ...
11
votes
1answer
785 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 ...
15
votes
4answers
3k 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?
24
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 ...
14
votes
4answers
556 views

Conjugate subgroup strictly contained in the initial subgroup?

Probably a very stupid question: Let $G$ be a group, $H\subset G$ a subgroup, $a\in G$ an element. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$? If $H$ has finite index or finite ...
6
votes
2answers
1k views

Finite subgroups of the multiplicative group of a field are cyclic

In Grove's book Algebra, Proposition 3.7 at page 94 is the following If $G$ is a finite subgroup of the multiplicative group $F^*$ of a field $F$, then $G$ is cyclic. He starts the proof by ...
25
votes
3answers
3k 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 ...
6
votes
3answers
752 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, ...
7
votes
2answers
466 views

The Center of $\operatorname{GL}(n,k)$

The given question: Let $k$ be a field and $n \in \mathbb{N}$. Show that the centre of $\operatorname{GL}(n, k)$ is $\lbrace\lambda I\mid λ ∈ k^∗\rbrace$. I have spent a while trying to prove this ...
7
votes
3answers
3k 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 ...
44
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 ...
17
votes
5answers
4k 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 $n$? ...
4
votes
2answers
197 views

An epimorphism from $S_{4}$ to $S_{3}$ having the kernel isomorphic to Klein four-group

Exercise $7$, page 51 from Hungerford's book Algebra. Show that $N=\{(1),(12)(34), (13)(24),(14)(23)\}$ is a normal subgroup of $S_{4}$ contained in $A_{4}$ such that $S_{4}/N\cong S_{3}$ and ...
2
votes
1answer
2k views

Subgroup of index 2 is Normal

Please excuse the selfishness of the following question: Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal. Proof: Because $|G:H|=2$, $G = H \cup aH$ for some $a \in ...
21
votes
5answers
2k 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 ...
7
votes
3answers
841 views

Problem from Herstein on group theory

The problem is: If $ G $ is a finite group with order not divisible by $ 3 $, and $ (ab)^{3} = a^{3} b^{3} $ for all $ a,b \in G $, then show that $ G $ is abelian. I have been trying this for a ...
7
votes
3answers
665 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 $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
14
votes
1answer
646 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. ...
1
vote
1answer
783 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
5answers
346 views

If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?

$P$ and $Q$ are subgroups of a group $G$. How can we prove that $P\cap Q$ is a subgroup of $G$? Is $P \cup Q$ a subgroup of $G$? Reference: Fraleigh p. 59 Question 5.54 in A First Course in ...
1
vote
2answers
190 views

Showing that $G$ is a group under an alternative operation.

Let $G$ be a group and let $c$ be a fixed elements of $G$. Now, I'm going to define a new operation "*" on $G$ by $a*b=ac^{-1}b$ How do I prove that the set $G$ is a group under *. Thanks for ...
25
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 ...
11
votes
3answers
1k views

No simple group of order 300

So Ive been trying to prove that there's no simple group of order 300. This is what I did and I was wondering if it was enough. Let |$G$|=2$^2$ . 3 . 5$^2$. Suppose $G$ is simple. Then there would be ...
13
votes
2answers
674 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 ...
11
votes
3answers
280 views

If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.

Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|>\frac{3|G|}{4}$, then $G$ is an abelian group. ...
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 ...
5
votes
4answers
966 views

Order of a product of subgroups

Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. I need this theorem to prove something.
14
votes
1answer
578 views

Finite group with isomorphic normal subgroups and non-isomorphic quotients?

I know it is possible for a group $G$ to have normal subgroups $H, K$, such that $H\cong K$ but $G/H\not\cong G/K$, but I couldn't think of any examples with $G$ finite. What is an illustrative ...
7
votes
3answers
2k 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.
5
votes
1answer
721 views

If a group is the union of two subgroups, is one subgroup the group itself?

"Let $G$ be a group, and suppose $G=H \cup K$, where $H$ and $K$ are subgroups. Show that either $H=G$ or $K=G$." Let $h \in H$ and $k \in K$. Then $hk \in H$ or $hk \in K$ (since every element of ...
62
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? ...
14
votes
2answers
2k 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 ...
15
votes
7answers
1k 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, ...
9
votes
12answers
5k views

Introductory Group theory textbook

What textbook is good for introductory group theory?
19
votes
2answers
515 views

Is every group the automorphism group of a group?

Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$? EDIT: It has been pointed out that the answer ...
4
votes
3answers
3k views

Right identity and Right inverse implies a group

Let $(G, *)$ be a semi-group. Suppose $ \exists e \in G$ such that $\forall a \in G,\ ae = a$; $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$. How can we prove that $(G,*)$ is ...
11
votes
2answers
2k 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 $G/H$. ...
10
votes
4answers
477 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 ...
4
votes
2answers
972 views

Nonabelian semidirect products of order $pq$?

I just constructed the semidirect product in Lang, and I'm trying to tie some facts together. From Ash's Algebra, I know that if $p\lt q$ are distinct primes, if $q\not\equiv 1\pmod{p}$, then any ...
12
votes
4answers
1k 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 ...
6
votes
1answer
182 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 ...
4
votes
2answers
517 views

Product of elements of a finite abelian group

Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$. Prove that if there is more than one element of order $2$ then $x=e$. What I've done so far: (#1 is just for ...
4
votes
3answers
1k views

Finite Index of Subgroup of Subgroup

Prove the following: If $H$ is a subgroup of finite index in a group $G$, and $K$ is a subgroup of $G$ containing $H$, then $K$ is of finite index in $G$ and $[G:H] = [G:K][K:H]$. So this is ...
10
votes
1answer
426 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.
18
votes
2answers
1k views

Finite number of subgroups $\Rightarrow$ finite group

I'm trying to prove that any group $G$ of infinite order has an infinite number of subgroups. I think that if the group has an element of infinite order, then it's easy because I can take the groups ...
4
votes
3answers
145 views

Needing help picturing the group idea.

I have a question; X and Y are subgroups of a group G. If $|X|, |Y| < \infty$, show that $|XY| = \frac{|X||Y|}{|X \cap Y|}$ but I can't really picture what it is talking about to even get ...
4
votes
3answers
5k 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 $G$.