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 $G\times K\cong H\times K$ imply $G\cong H$?

Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?
7
votes
1answer
227 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 ...
9
votes
4answers
2k views

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 ...
5
votes
3answers
3k views

Does the intersection of two finite index subgroups have finite index?

Let $(G,*)$ be a group and $H,K$ be two subgroups of $G$ of finite index (the number of left cosets of $H$ and $K$ in $G$). Is the set $H\cap K$ also a subgroup of finite index? I feel like need that ...
2
votes
1answer
294 views

Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.

I was thinking that the product of groups $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2\mathbb{Z}$ is not cyclic, but $\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/p\mathbb{Z}$ is cyclic if p is an odd prime. ...
1
vote
3answers
168 views

An example of a residually finite group which is not Hopf

trying to think of any residually finite group which is not Hopf. Any help?
186
votes
28answers
17k 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 ...
70
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? ...
59
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 ...
16
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 ...
17
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14answers
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17
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7answers
5k 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 ...
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, ...
13
votes
2answers
1k views

Theorems with the greatest impact on group theory as a whole

In his Contemporary Abstract Algebra text, Gallian asserts that Sylow's Theorem(s) and Lagrange's Theorem are the two most important results in finite group theory. He also provides this quote by ...
19
votes
2answers
750 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 ...
9
votes
4answers
3k views

Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ ...
8
votes
2answers
1k 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 ...
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 ...
9
votes
4answers
369 views

If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?

Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$? Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using ...
2
votes
2answers
297 views

Isomorphic Group with $G=(\mathbb Z_{2^\infty}\oplus \frac{\mathbb Q}{\mathbb Z}\oplus \mathbb Q)\otimes_{\mathbb Z}\mathbb Q $

Let $$G=\left(\mathbb Z_{2^\infty}\oplus\mathbb Q/\mathbb Z\oplus \mathbb Q\right)\otimes_{\mathbb Z}\mathbb Q $$ Now $G$ isomorphic with which case: $0$ ? , or $\mathbb Q \, $ ? , or $\mathbb ...
7
votes
2answers
3k views

Show that every group of prime order is cyclic

Show that every group of prime order is cyclic. I was given this problem for homework and I am not sure where to start. I know a solution using Lagrange's theorem, but we have not proven ...
5
votes
2answers
1k 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 ...
11
votes
2answers
2k views

Group of positive rationals under multiplication not isomorphic to group of rationals

A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{>0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} ...
7
votes
1answer
750 views

Infinite group with only two conjugacy classes

Can you show me a reasonably simple (using only elementary group-theoretic tools) example of infinite group with just 2 conjugacy classes ?
2
votes
3answers
603 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 ...
3
votes
3answers
765 views

Image of subgroup and Kernel of homomorphism form subgroups

Is my proof ok? Let $f:G\to G^{\prime}$ be a group homomorphism and let $H\lt G$. $Im(H) = \{f(x):x\in H\}$. To show that $Im(H)$ is a group, it suffices to show that $f(x)f(y)^{-1}\in Im(H)$. ...
3
votes
3answers
293 views

Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...
2
votes
3answers
287 views

Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$ [duplicate]

Another homework problem from my Group Theory class. Let $a,b$ be elements of a group, $G$. Let $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. Let $a$ and $b$ commute. Prove: If $m$ and ...
6
votes
3answers
7k 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 $G$.
4
votes
3answers
152 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 ...
2
votes
2answers
235 views

When does the isomorphism $G\simeq ker(\phi)\times im(\phi)$? hold?

Suppose you have a group isomorphism given by the first isomorphism theorem: $G/ker(\phi) \simeq im(\phi)$ What can we say about the group $ker(\phi)\times im(\phi)$? In particular, when does the ...
48
votes
3answers
5k 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 ...
26
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 ...
22
votes
1answer
695 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 ...
9
votes
1answer
395 views

Involutions and Abelian Groups

Suppose that $ G $ is a finite group where at least three-fourths of the elements are involutions, i.e., $$ |I(G)| \geq \frac{3}{4} |G|. $$ (Here, $ I(G) $ denotes the set of all involutions of $ G $, ...
23
votes
1answer
561 views

“Efficient version” of Cayley's Theorem in Group Theory

I'm considering finite groups only. Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$. I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
17
votes
1answer
481 views

A special subgroup of groups of order $n$

Let $G$ be a group with $|G| = n$ and let $ \emptyset \ne S \subseteq G$. I want to show that $S^n$ is a SUBGROUP of $G$ where by $S^n$ I mean the set $\lbrace s_1\cdots s_n \; | \; s_i \in S\rbrace$. ...
15
votes
2answers
698 views

Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?

This Wikipedia article states that the isomorphism type of a finite simple group is determined by its order, except that: $L_4(2)$ and $L_3(4)$ both have order $20160$ $O_{2n+1}(q)$ and $S_{2n}(q)$ ...
14
votes
5answers
3k views

How can you show there are only 2 nonabelian groups of order 8?

It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. I've never ...
8
votes
2answers
686 views

What is the center of a semidirect product?

Let $G_1$ and $G_2$ be groups. Let $\varphi:G_2\rightarrow \operatorname{Aut}(G_1) $ be a group homomorphism defining the semidirect product $G_1 \rtimes G_2$. Determine the center ...
12
votes
1answer
213 views

Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
12
votes
1answer
516 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 ...
17
votes
4answers
6k views

Why are two permutations conjugate iff they have the same cycle structure?

I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
7
votes
2answers
1k views

If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$

I'm struggling to proof that if $H$ and $K$ are subgroups of a finite index of a group $G$ such that [G:H] and [G:K] are relatively prime, then $G=HK$. I don't know why I can't answer it, because this ...
19
votes
1answer
818 views

Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?

Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups? I know that there is a bijection between $\mathbb{R}$ and $\mathbb{C}$, and this question asks whether they are isomorphic as ...
15
votes
5answers
2k views

Prove that this is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, ...
15
votes
4answers
2k 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
2answers
368 views

Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
10
votes
5answers
2k views

Why is the set of commutators not a subgroup?

I was surprised to see that one talks about the subgroup generated by the commutators, because I thought the commutators would form a subgroup. Some research told me that it's because commutators are ...
9
votes
2answers
3k views

Are normal subgroups transitive?

Suppose $G$ is a group and $K\lhd H\lhd G$ are normal subgroups of $G$. Is $K$ a normal subgroup of $G$, i.e. $K\lhd G$? If not, what extra conditions on $G$ or $H$ make this possible? Applying the ...