A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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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 ...
25
votes
4answers
834 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 ...
50
votes
1answer
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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 ...
31
votes
4answers
4k 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 ...
17
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5answers
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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. ...
22
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5answers
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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
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2answers
3k 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 ...
18
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1answer
1k 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 $2n$, show that the number of elements of $G$ of order $2$ is odd. That is, for some integer $k$, there are ...
12
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1answer
1k 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 ...
12
votes
3answers
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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 ...
13
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4answers
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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 ...
6
votes
2answers
280 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 ...
7
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3answers
1k 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 ...
6
votes
1answer
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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 ...
8
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4answers
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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, ...
29
votes
5answers
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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 $n$? ...
31
votes
3answers
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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 ...
9
votes
3answers
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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 ...
5
votes
1answer
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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 ...
54
votes
2answers
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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 ...
14
votes
5answers
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Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems?
9
votes
5answers
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Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime

How can I show that $(n-1)!$ is congruent to $-1 \pmod{n}$ if and only if $n$ is prime? Thanks.
5
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3answers
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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 ...
6
votes
2answers
742 views

Normal subgroups of $S_4$

Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$? In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?
12
votes
3answers
433 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. ...
6
votes
3answers
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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 ...
14
votes
1answer
733 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 ...
12
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4answers
853 views

Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions

I have some difficulties with a task in algebra. I guess it's trivial and really easy but I can't figure out how to solve it. I have a set $G$ and a binary operation on it, let it be $\circ$. I have ...
2
votes
1answer
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Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\nmid n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
6
votes
5answers
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Subgroups of a cyclic group and their order.

Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states, Let $G = \langle a \rangle$ be a cyclic group. (i) Every subgroup $S$ of $G$ is cyclic. (ii) If ...
1
vote
2answers
307 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 ...
22
votes
5answers
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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 ...
17
votes
2answers
584 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 ...
11
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3answers
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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.
8
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3answers
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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 $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
19
votes
1answer
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Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?

I think the answer is yes. Sketch of the proof Consider $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Let $\{e_\lambda:\lambda\in\Lambda\}\subset\mathbb{R}$ be its Hamel basis. Then ...
16
votes
3answers
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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 ...
11
votes
4answers
616 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 ...
7
votes
2answers
926 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 ...
2
votes
3answers
638 views

Cancellation of Direct Products

Given a finite group $G$ and its subgroups $H,K$ such that $$G \times H \cong G \times K$$ does it imply that $H=K$. Clearly, one can see that this doesn't work out for all subgroups. Is there any ...
11
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1answer
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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.
10
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4answers
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Is it true that the order of $ab$ is always equal to the order of $ba$?

How do I prove that if $a$, $b$ are elements of group, then $o(ab) = o(ba)$? For some reason I end up doing the proof for abelian(ness?), i.e., I assume that the order of $ab$ is $2$ and do the ...
3
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5answers
414 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 ...
196
votes
28answers
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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 ...
33
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4answers
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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 ...
73
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2answers
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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? ...
61
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8answers
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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 ...
31
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4answers
2k 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?
13
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3answers
2k views

No simple group of order $300$

So I've 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. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there ...
14
votes
1answer
496 views

Does $gHg^{-1}\subseteq H$ imply $gHg^{-1}= H$? [duplicate]

Let $G$ be a group, $H<G$ a subgroup and $g$ an element of $G$. Let $\lambda_g$ denote the inner automorphism which maps $x$ to $gxg^{-1}$. I wonder if $H$ can be mapped to a proper subgroup of ...