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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$\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
2
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1answer
37 views

$R$ be a commutative unital ring , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+) \ncong (R^{\times} , .)$?

Let $R$ be a commutative ring with unity , let $R^{\times}$ be the group of units of $R$ , then is it true that $(R,+)$ and $(R^{\times} ,\cdot)$ are not isomorphic as groups ? I know that the ...
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A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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1answer
25 views

Homomorphism between S4 and A4

I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H. I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 ...
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1answer
57 views

find torsion coefficients of groups

I have to find torsion coefficients of groups $G_1\simeq Z/2\oplus Z/4\oplus Z/3\oplus Z/3\oplus Z/9$ and $G_2\simeq Z/15\oplus Z/20\oplus Z/18$. I want to ask if my calculations are correct. For $...
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52 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
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1answer
21 views

possible cyclic group from fundamental theorem of finite abelian

Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}...
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17 views

Pronormal subgroups of direct products

Suppose that $G = A \times B$. Let $U = A \times \pi_B(U) \leq G$ such that $\pi_B(U)$ is pronormal in $B$. Then $U$ is pronormal in $G$. This is part of a proof of Proposition 4.3 in Pronormal ...
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14 views

Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
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25 views

Isomorphism between semidirect products

Alperin and Bell, Groups and Representations, section 2, Proposition 11, p. 23, states: "Let $H$ be a cyclic group and let $N$ be an arbitrary group. If $\varphi$ and $\psi$ are monomorphisms from $H$ ...
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3answers
45 views

Does the converse of Lagrange's theorem hold for any finite Dedekind group?

I'm currently reading a book on algebra and the following argument is used to prove that the converse of Lagrange's theorem (if $d$ divides $|G|$ there exists a subgroup of order $d$) holds for any ...
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3answers
57 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
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2answers
24 views

Normal closure of a subgroup of a free group.

Let $G$ be a finitely generated free (nonabelian) group, $H$ a subgroup generated by some of the generators of $G$, and $a: G\to AG$ be the projection to the abelianization $AG:=G/[G,G]$. Is it true ...
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0answers
32 views

Question about generators and Hom functor

In a given category $\mathcal{C}$ I want to prove the following statement: If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{...
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0answers
21 views

How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
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4answers
57 views

If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
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0answers
44 views

Error Correcting Code and Graph Theory

I am currently in an introductory graph theory class, and we are supposed to give a short presentation by the end of the semester. Recently, I've learned (a very small amount) about error correcting ...
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4answers
618 views

Problems from the Kourovka Notebook that undergraduate students can fully appreciate

The Kourovka Notebook is a collection of open problems in Group Theory. My question is: could you point out some (a "big-list" of) problems [by referencing them] presented in this book that are, ...
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4answers
53 views

$\mathbb{C}/\mathbb{Z}$ is isomorphic to multiplicative group $\mathbb{C}\setminus\{0\}$ [duplicate]

I have to show that $\mathbb{C}/\mathbb{Z}$ is isomorphic to the multiplicative group $\mathbb{C} \setminus \{0\}$. Proof. Let $f:\mathbb{C} \setminus \{0\} \rightarrow \mathbb{C}/\mathbb{Z}$ be the ...
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2answers
50 views

How can I find all the numbers of order 10 in $Z_{60}$? [on hold]

How can I find all the numbers of order $10$ in $Z_{60}$? In fact, these are all the numbers with $(x,60)=6$. How can I find all these numbers?
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1answer
43 views

When is $(\Bbb Z/n\Bbb Z)^\times$ cyclic? [duplicate]

Is the group of units $(\Bbb Z/n\Bbb Z)^\times$ always cyclic? Do we need that $n$ is a prime or something?
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1answer
22 views

All homomorphisms from $Z/4Z$ to $Z/6Z$

I am asked to find all group homomorphisms from $Z/4Z$ to $Z/6Z$. Let $f:Z/4Z \rightarrow Z/6Z$ be such a homomorphism. By definition we have $f(1) = 1$ and therefore $f(0)=f(1 * 0) = f(1) * f(0) = ...
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2answers
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order of an element in a modulo group under multiplication

Suppose $G$ is the group $ℤ_{37}^\times$ under multiplication. Then is there a way that I can prove the order of the element $2$ in $G$ is $36$ without finding all the powers of $2$ until I get unity?...
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1answer
89 views

Finding permutation with a condition

Let $a$ be a permutation in $S_6$. I'm asked whether there is an $a$ so that $a^2 = (123)(456)$ I'm quite confused about where to start. I do know the $a$ must consist of $3$ elements (right?). How ...
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2answers
178 views

About an article regarding free groups

I am currently reading this paper https://blms.oxfordjournals.org/content/35/5/624.abstract and I have some difficulties to understand two steps of the proof of the main theorem. Let $G$ be a non ...
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3answers
16 views

Sylow $p$-subgroups in $S_p$ and order $p$ elements

I realise this question has been asked a few times before, but I don't understand the answers. What is the number of Sylow p subgroups in S_p? Why are the order $p$ elements in $S_p$ exactly cycles ...
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0answers
17 views

Direct product of join of subgroups

Suppose that $G$ is a finite group with $A, B, H, K \leq G$. Suppose that $H\times A \leq G$ and $K\times B \leq G$. I want to show that $\langle H,K \rangle \times \langle A, B \rangle = \langle H \...
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1answer
74 views

What is the number of Sylow p subgroups in S_p?

I am reading the Wikipedia article entitiled Sylow theorems. This short segment of the article reads: Part of Wilson's theorem states that (p-1)! is congruent to -1 (mod p) for every prime p. One ...
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2answers
55 views

Let $G$ be a (probably infinite) group and $H \leq K \leq G$. $|G:H| < \infty \Rightarrow |G:H|=|G:K||K:H|$?

Let $G$ be a (probably infinite) group and $H \leq K \leq G$. Is it true that if we have $|G:H| < \infty$ then $|G:H|=|G:K||K:H|$ ? Thanks in advance.
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1answer
25 views

If $K\leq H\leq G$ (not necessarily finite groups). Then prove that $[G:K]=[G:H]\cdot [H:K]$ [duplicate]

Let $K\leq H\leq G$ (not necessarily finite groups). Why do we have $[G:K]=[G:H]\cdot [H:K]$? I can't figure out a proof in the setting of possibly infinite groups and non-normal subgroups.
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2answers
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Let $G$ be finite group and $H<G$. For every proper subgroup $K$, $[G:H]\leq[H:K]$.

Let $G$ be finite group and $H<G$. For every proper subgroup $K$, $[G:H]\leq[H:K]$. I want to prove $H$ is normal subgroup. I fixed $K:=g^{-1}Hg$ but this doesn't work. Can somebody advise me?
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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 ...
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0answers
36 views

Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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2answers
47 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 ...
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0answers
30 views

Can someone find example for it? [on hold]

Example for group that for any element the order is $2$ apart of $e$. (without $Z_2+Z_2+Z_2+\cdots+Z_2$)
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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
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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 ...
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2answers
33 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 ...
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0answers
34 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 ...
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1answer
26 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 $n$, ...
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1answer
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How to explain this contradiction about Weyl group of $SL_n(K)$?

I have some difficulties in understanding why the Weyl group of algebraic group $SL_n(K)$ is isomorphic to symmetric group $S_n$. Let $G=SL_n(K)$ be the simply-connected algebraic group over the ...
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1answer
417 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 ...
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37 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 ...
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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 ...
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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 $x\in M$, ...
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2answers
26 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 ...
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6answers
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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 $n$? ...
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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 ...
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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
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5answers
510 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 $m$ ...