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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2
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2answers
57 views

Normal subgroups of $A_5$ must contain a 3-cycle.

I am trying to prove the simplicity of $A_5$ by showing that every non-trivial normal subgroup $H$ contains a 3-cycle, and therefore is all of $A_5$ since the 3-cycles all belong to one conjugacy ...
0
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2answers
48 views

Non-abelian Order of $6$ is isomorphic to $S_3$ [on hold]

I know that it's duplicate, but , How can I prove it? I know that must be element "$a$" of order 2, and element "$b$" of order 3. What is the next step? In fact, from what I search it is claimed that ...
-3
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0answers
31 views

Can someone find example for it? [on hold]

Example for group that for any element the order is $2$ apart of $e$. (without $Z_2+Z_2+Z_2+\cdots+Z_2$)
-3
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0answers
25 views

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? [on hold]

It's True? For any group G that his order is pq {p,q primes} has 2 nontrivivallic normal subgroups? If yes, so hwo can i prove it? [without Sylow Theorems] p,q > 1
8
votes
1answer
160 views

Does this particular axiom on a semigroup guarantee that it is a group?

Update: Eric Wofsey has demonstrated the conjecture in the commutative case below, and Tobias Kildetoft has provided a simple counterexample to the non-commutative claim. This would-be replacement ...
0
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2answers
34 views

$G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
1
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0answers
37 views

Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
0
votes
1answer
28 views

Linear representations of projective groups

Does the projective linear group $PSL_2(\mathbb{R})$ admit faithful linear representations? In other words, does there there exist a homomorphism $SL_2(\mathbb{R}) \to GL_n(\mathbb{R}),$ for some $n$, ...
6
votes
1answer
418 views

Proving that a group of order $pqr$ (with conditions on those primes) is abelian.

I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ...
0
votes
2answers
38 views

Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
1
vote
1answer
9 views

number of orbits by action of $D_{12}$ on $\mathbb{Z}_{12}^k$

Let $X=\mathbb{Z}_{12}^k$ for $k\in \mathbb{N}$ and $G=D_{12}$. Define an action of $D_{12}$ on $X$ by setting rotations $r^n(p)=(p_1+n,\dotsc,p_k+n)$ where the coordinates are taken modulo $12$ and ...
1
vote
0answers
28 views

Is power-associativity an equational property?

A magma $M$ is said to be power-associative if the subalgebra generated by any element is associative. This can be written simply as $x^mx^n=x^{m+n}$ for all $m,n$ positive integers and $x\in M$, ...
0
votes
2answers
28 views

Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
30
votes
6answers
8k 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 $n$? ...
8
votes
2answers
803 views

Abelian Group Element Orders [duplicate]

I want to show that if a finite abelian group has elements of order $m$ and $n$ then it will have an element of order $\text{lcm}(m,n)$. First I proved the lemma if $a$ has order $m$ and $b$ has ...
14
votes
3answers
389 views

A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?

Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$ Then how prove $G$ is an abelian ...
2
votes
5answers
512 views

A criterion for a group to be abelian [duplicate]

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let $m$ ...
3
votes
1answer
30 views

Problem with proof of $H \cap K $ is of finite index if $ H,K$ are finite index subgroups

I came across a proof earlier for a solution to a Herstein Topics in Algebra question earlier that I'm not convinced with, from AOPS site. If $G$ is a group and $H,K$ are two subgroups of finite ...
1
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4answers
676 views

Groups of order 12 that aren't isomorphic [duplicate]

Give examples of four groups of order 12 no two of which are isomorphic. So far I've thought of $Z_{12}$ and $D_6$. Thanks!
-1
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1answer
30 views

The sum of a family of submodules

I'm just reading Atiyah-Macdonald's commutative algebra text, specifically the section on Operations on Submodules. I just have some questions regarding constructions. Firstly, suppose $M$ is an $A$-...
1
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1answer
21 views

Conjugation of a direct product

Suppose that $H \times K \leq G = A\times B$ and let $g\in G$. Is it true that $g( H\times K)g^{-1} = gHg^{-1} \times gKg^{-1}$ holds in general?
1
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4answers
51 views

is the given structure a group

is the below given structure a group?? [ * e a b c d f g ...
2
votes
1answer
47 views

Kernel of a group action of rotation of a cube

Question: Let G be the rotation group of a cube Show that G has an action on a set of size 3. Well, if we consider axes through each opposite faces, then this set has only 3 possible axes. ...
0
votes
1answer
48 views

Kernel of homomorphism is normal subgroup

When trying to prove that a normal subgroup is a kernel, you map x to Nx(the coset). f(x) = Nx Why exactly do we do that? Why when you take an element in the normal ...
1
vote
1answer
38 views

Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
0
votes
0answers
25 views

Is this set a subgroup

I was reading lemma 3 of this proof. Is $P^p$ a subgroup ? We know that for any abelian group $G$, the set $G^n$ is a subgroup. But is it true for $P^p$ ?
1
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1answer
49 views

Elements that are their own inverses in a symmetric group.

How many elements are their own inverses in $S_6$? I'm having a hard time figuring out how to calculate such a thing.
-1
votes
2answers
53 views

Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
-1
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1answer
48 views

Not Abelian group G with Z(G) that contains only two elements? [on hold]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
1
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3answers
36 views

Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
0
votes
1answer
43 views

interchanges/transpositions (how to read)

I have came across this before and just again now, in the same form of which I'm struggling to understand. Although I know it's link to parity, as a perm group pi: $$ \pi = \begin{pmatrix} 0 & 1 ...
1
vote
0answers
41 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
2
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0answers
44 views

Homeomorphisms of the circle

I know that there is a vaste litterature about the group of the homeomorphisms of the circle. I would a good reference to start the study of this topic. Thanks in advance.
-1
votes
2answers
31 views

Meaning of operation preservation in group homomorphism [on hold]

What is meaning of operation preservation in a group homomorphism from a group $G_1$ to a group $G_2$?
1
vote
1answer
47 views

About the generators of a free group

Suppose we know that $G$ is a free group of rank $n$ and that $\{g_1,...,g_n\}$, with all the $g_i$ distinct, is a system of generators for $G$. Are we sure that it is also a $\textbf{free}$ system of ...
1
vote
0answers
28 views

cardinality of a maximal subgroup of a $p$ group

Let $P$ be a $p$ group with $|P|=p^n$. Let $M$ be a maximal subgroup of $P$. Is it true that $|M|=p^{n-1 }$ ?
1
vote
1answer
32 views

A sufficient condition for profinite groups

I know that Edwin Hewitt and Kenneth A. Ross (1970) show: Any compact Hausdorff torsion group is profinite. But I don't have the book, the proof seems long and I need only the case of abelian groups ...
1
vote
1answer
31 views

how to check if a subgroup is maximal?

Is there any general strategy to check whether a subgroup is maximal or not ? For example, in case of rings, we know that an ideal $I$ of a ring $R$ is maximal if and only if $R/I$ is a field. Is ...
4
votes
1answer
48 views

Prove that up-to isomorphism there are two integral domains of order $p^2$.

Prove that up-to isomorphism there is exactly one integral domain of order $p^2$ . Does there exist only two non-commutative rings of order $p^2$ upto isomorphism? We know that any group of ...
3
votes
1answer
35 views

Existence of open subgroup extending a smaller one

Let $G$ be an abelian topological group and $H \subseteq G$ a dense subgroup (equipped with the subset topology). Furthermore let $V \subseteq H$ be a subgroup that is open in $H$. Does there exist a ...
0
votes
1answer
57 views

Need help in understanding the example from Dummit & Foote text

This is an example from Dummit & Foote text. Let $D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>$.Since [$r,s$]$=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
0
votes
1answer
35 views

Group Theory: Orders of Elements

Have been given following problem. Let G be group and x $\in G$. Prove that $x^2 = e$ if and only if x is of order one or two. My response. If order of x is 1, then by definition of order of an ...
7
votes
1answer
551 views

groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
24
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3answers
3k views

Why isn't the orbit-stabilizer theorem obvious?

The title of this post paraphrases the title of a great blog post by Timothy Gowers, where he argues that those who think that the fundamental theorem of arithmetic is obvious are almost certainly ...
1
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1answer
25 views

Prove all products of two disjoint 2-cycles are pairwise conjugate in $A_{n}$

Let $n \geq 4$. Prove that in the group $A_{n}$ of parity-even permutations of $\{1,2,...,n\}$, all products of two disjoint 2-cycles are pairwise conjugate. This is a past exam question for a ...
1
vote
2answers
65 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
0
votes
3answers
47 views

Determining all the elements of S4?

What is an easy way to determine the elements of $S_{4}$? While going through my revision process in an attempt to stem out the nitty gritty areas that I am unsure of, I chanced upon this. I tried ...
0
votes
0answers
29 views

All 3-cycles are pairwise conjugate.

In my lectures on Group Theory our lecturer claimed the following: All 3-cycles are pairwise conjugate. He then went on to prove this but I am struggling with understanding his proof. I will try and ...
3
votes
3answers
391 views

Sylow p-subgroup of a direct product is product of Sylow p-subgroups of factors

Let $G$=$HK$ be a finite group (direct product), $P$ a Sylow $p$-subgroup of $G$. Prove that $P$ = $H'K'$, where $H'$ and $K'$ are Sylow $p$-subgroups of $H$ and $K$ respectively. I am very new to ...
2
votes
1answer
48 views

how to calculate dimension of quotient space?

what is the dimension of $O(n) \setminus \mathbb{R}^n$ for $n \geq 2$? how to calculate this? $n - \frac{n(n-1)}{2}$ does not work well.