Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Elements of symmetric groups

How to prove that (123) is not a cube of any element in $S_n$. Is it true in general that any $p$ cycle, $p$ an odd prime, can't be written as a $p^{th}$ power of any element in $S_n$? Thanks.
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1answer
50 views

G to H is a homomorphism, o(h) = 100, what are the values of o(g)?

Question: Let $\phi:G \rightarrow H$ be a homomorphism and let $g\in G$ and set $h=\phi(g)$. Suppose $o(h)=100$. Assume $g$ has finite order. What are the possible values of $o(g)$? Attempt: ...
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1answer
88 views

Classification of finite simple AC groups.

A group is called an AC-group if the centralizer of every non-central element is abelian. So far i have known only one class of groups which is a finite AC simple group, namely the group $PSL(2,q)$, ...
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1answer
509 views

Classify Finite Abelian Groups of Order 8

Without using the fundamental theorem of finite abelian groups, show that, if $G$ is a finite abelian group of order 8, then $G$ is isomorphic to one of $\mathbb{Z} / 8\mathbb{Z}$, $\mathbb{Z} / ...
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2answers
103 views

$(1 2)(3 4)$ does not commute with any nonidentity element of odd order in $A_5$.

On Dummit's Abstract Algebra on p. 128, it says: "It is easy to see that $(1 2)(3 4)$ ... does not commute with any non-identity element of odd order in $A_5$." But I don't find it easy. Any ...
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2answers
184 views

Why must the order of basepoint of elliptic curve be prime?

Let $E$ be an elliptic curve defined over a finite field $F(q)$. Let $G\in E(F(q))$ be a point of order $n$, where $n$ is a prime number and $n>2^{160}$. The elliptic curve discrete logarithm ...
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2answers
79 views

Solve question using this hint

This is not for homework, and I've shown the problem without using this hint before, but I am just trying to understand how the following hint is helpful. The problem asks: If $G$ is a finite group ...
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1answer
149 views

Does every $p$-group of odd order admit fixed point free automorphisms?

Does every $p$-group of odd order admit fixed point free automorphisms? equivalently, Given an odd order $p$-group $P$, is there a group $C$ such that we can form a Frobenius group $P\rtimes ...
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4answers
90 views

Isomorphism question about groups

I know that the group $\mathbb{Z}/9\mathbb{Z}$ is not isomorphic to the group $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$, but I just do not know how to prove this.
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2answers
101 views

Normal subgroup in a $p$-group [duplicate]

Let $G$ be a $p$-group, and $H$ is a normal subgroup of $G$ with $|H| = p$. Prove that $H \leq Z(G)$. More general, if $K$ is a normal subgroup of $G$, then $K\cap Z(G) \neq \{e\}$, with $e$ is the ...
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1answer
92 views

non-split extension and Schur multiplier

Let $G$ be a central extension of the group $K$ by the simple non-abelian group $H$ ($K$ is the normal subgroup). If we know that this extension is non-split, is it true that the order of $K$ must ...
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1answer
81 views

Make $3$-hypergraph Cayley on $ D_{2n} $ and $ \mathbb Z_n$

How we can make $3$-hypergraph Cayley on $ D_{2n} $ and $ \mathbb Z_n$? Definition: let $G$ be a group, A $3$-hypergraph cayley on $G$ has a generator set $T$ with elements of order $3$ such that ...
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2answers
187 views

Conjugate subgroups and conjugate elements

While trying to prove that the alternating group $A_5$ is a simple group, I came across two assertions I see as contradicting, that is : the 5-cycles are not all conjugate to each other (proven ...
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1answer
155 views

Is there a Schur-Zassenhaus-free proof that $\Phi(G)$ cannot contain a Sylow subgroup of $G$?

As we know, the Frattini subgroup of a finite group G can not contain a Sylow subgroup of G, but if we want to prove this, we need the Schur-Zassenhaus theorem. What I want to know is if there is a ...
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3answers
50 views

Show a group with the transpositions

Show that $S_4=\langle{(12),(1234)}\rangle$. These are the transpositions. should I start with all the groups of $S_4$ , $S_4=4!$ And go about in proving the cosets of $S_4$?
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1answer
63 views

A problem of permutation group

An exercise in a book of permutation groups: Let $G \leq S_n$. If $G$ has $r$ orbits, show that $G$ can be generated by a set of at most $n-r$ elements. I really have no idea how to prove it. ...
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1answer
74 views

Use Cauchy Theorem to prove that every element in a group $G$ is a $k$-th root iff $(k,|G|)=1$

Every element of a finite group $G$ has a $k$-th root if and only if $(k,|G|)= 1$. I want to prove this proposition, I´m trying to use this function that for all g in G , g will send it to $g^k$ ...
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3answers
79 views

Is this Cayley table correctly computed? If so, is it correct way of presentation of this dihedral group?

Is the following table for $D_4$ correct? $$\begin{array}{c|c|c|c|c|c|c|c|c|} * & 1 & g & g^2& g^3& f& fg & fg^2 & fg^3 \\ \hline 1 & 1 & g & g^2 ...
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0answers
97 views

The conjugacy classes of the simple group PSL(2,q)

If $q=p^{\alpha}$, where $p$ is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$ and $2$ in the simple group $PSL(2,q)$.
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2answers
240 views

Derived subgroup of a group whose all it's Sylow subgroups are cyclic is abelian. [closed]

Let $ G $ be a finite group such that all its Sylow subgroups are cyclic. Prove that $ G'$ is abelian. (Here $ G' $ denotes the derived subgroup.)
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1answer
55 views

How to find this formula in this dihedral group of transformations of the plane?

In the group of all the bijections of the Euclidean plane onto itself, let $f(x,y) \colon = (-x,y)$ and $g(x,y) \colon = (-y,x)$ for all points $(x,y)$ in the plane. Let $$G:= \{f^i g^j | i=0,1; \ ...
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1answer
48 views

what can we know about this kind of group

Let G be a finite group,H is an arbitrary proper subgroup of G,H is solvable,but G is not solvable.then what can we know about group G?
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1answer
63 views

Algebraic expressions and permutation groups

Suppose that I pick a subgroup $G$ of $S_n$ for some $n$. Is it always possible to find an algebraic expression in $n$ variables (in other words, a rational function in those $n$ variables) that is ...
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4answers
372 views

Use every non-abelian group of order 6 has a non-normal subgroup of order 2 to classify groups of order 6.

Prove that every non-abelian group of order $6$ has a non-normal subgroup of order $2$. Use this to classify groups of order $6$. I proved that every non-abelian group of order 6 has a nonnormal ...
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1answer
70 views

Are there asymptotically more nonabelian groups of order $p^k$ than there are abelian groups of order $\leq p^k$?

Let $\alpha(n)$ denote the number of isomorphism classes of abelian groups of order $n$ and $\alpha^\prime(n)=\sum_{m=1}^n\alpha(m)$. Similarly, define $f(p^k)$ to be the number of isomorphism ...
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1answer
90 views

Are there groups with conjugacy classes as large as the divisors of perfect numbers?

Are there groups, where the conjugacy classes have elements according to the divisors of perfect numbers larger than $6$? I already got the first example, so therefore $6$ is excluded: $S_3$. The ...
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3answers
142 views

Which elements could possibly commute with a cycle of full length in $S_n$?

In the symmetric group of degree $n$, which elements could possibly commute with the permutation $\sigma$ given by $\sigma(i) = i+1$ if $i < n$; $\sigma(n) = 1$? Of course, the permutations $e= ...
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1answer
46 views

Are groups of component type always of Lie type, alternating or sporadic?

In http://en.wikipedia.org/wiki/Classification_of_finite_simple_groups it was written that "A group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component ...
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1answer
113 views

Cauchy-Davenport theorem and its extension

According to Cauchy-Davenport Theorem, if $A,B$ are subsets of a prime field ($F_p$) then we have the following bound on the number of elements within the sumset $A + B = \left\{ {\left. {a + b} ...
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1answer
130 views

How can I explain the sketch of the proof of classification of finite simple groups to someone who knows only basics of algebra?

This question was popped up in my mind when I read Finnish Wikipedia. How can I explain the sketch of the proof to layman? Is it worth to explain for example Ree groups in the text or just say ...
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1answer
34 views

if $G$ is a EDP of two finite groups A and B. then order of element $(a,b)\in G=A\times B$ is lcm of order of a and order of b.

If $G$ is a external direct product of two finite groups $A$ and $B$, then order of element $(a,b)\in G=A\times B$ is lcm of order of $a$ and order of $b$.
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1answer
97 views

Finite group in $GL(n,\mathbb{Q})$ is conjugate to finite group in $GL(n,\mathbb{Z})$? [duplicate]

I can't solve this problem: Finite group in $GL(n,\mathbb{Q})$ is conjugate to finite group in $GL(n,\mathbb{Z})$. Could any one help me? Thanks a lot!
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2answers
91 views

How to prove this assertion in $S_n$ for $n \geq 3$?

Let $n \geq 3$. Then there exists an element $f \in S_n$ such that $f \neq g^3$ for any element $g \in S_n$, where $S_n$ denotes the symmetric group on $n$ letters. How to establish whether this ...
3
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1answer
67 views

Is there an automorphism of symmetric group of degree 6 sending a transposition to product of two transpositions?

$\operatorname{Aut}(S_6)\cong S_6\rtimes C_2$. there are several (720) automorphisms sending a transposition to product of three transpositions. Is there an automorphism sending a transposition to ...
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3answers
118 views

What's the smallest exponent to give the identity in $S_n$?

Let $S_n$ denote the symmetric group on $n$ letters. We know that $\tau^{n!} = e$ for any element $\tau \in S_n,$ where $e$ denotes the identity element. Can we find a smaller positive integer $m$ ...
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2answers
151 views

Composition series and chief series of $p$-group

composition series and chief series of $p$-group. How to solve the following Problem? Thanks. Let $G$ be a group of order $p^n$, $p$ prime. Prove every chief factor and every composition factor is of ...
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0answers
73 views

Extensions of elementary abelian p-groups by themselves.

How many distinct extensions of $C^k_p$ by $C^l_p$ are there where $C^n_p$ is elementary abelian of order $p^n$, p a prime?
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2answers
102 views

Find a chief series for dihedral group $D_{2n}$

The question is : Find a chief series for dihedral group $D_{2n}$. Is each normal subgroup of $C_n$ normal in $D_{2n}$?
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1answer
397 views

Cayley table for a group of order 5

I know how to work the cayley table up to a group of order 4. I'm having difficulty in doing one with order 5
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1answer
94 views

$\operatorname{Aut}(G)$ contains an involution $\sigma$ with no nontrivial fixed point

I am just reading some algebra books on my own, and it seems the following exercise appears in so many of them: Let $G$ be a finite group with $\sigma\in\operatorname{Aut}(G)$ satisfying ...
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0answers
40 views

Determining all subgroups of $S_3$ [duplicate]

Related to : How would I prove what elements $S_3$ contains, and what its subgroups are? I don't know Lagrange's theorem, and I don't want to prove that there are no subgroups of size $4$ or $5$, is ...
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2answers
154 views

How would I prove what elements $S_3$ contains, and what its subgroups are?

Given that $S_3$ is a symmetric group of size three, how would I find all elements of it, and all subgroups?
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2answers
227 views

Group of order 96. Show there exists a normal subgroup of order 16 or 32.

Let $G$ be a group. Show that there exists a normal subgroup of $G$ such that its order is either 16 or 32. Attempt at a solution: If $n_{2}=1$ (the number of Sylow 2-subgroup of G) then we have a ...
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3answers
140 views

Maximal abelian subgroups in a $p$-group are always normal?

Does all the maximal abelian subgroups in a given $p$-group have to be normal? I doubt the correctness of this claim however couldn't manage to find a counterexample. Can anyone give me some advice? ...
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1answer
76 views

Counting the subgroups of $\Bbb Z_4 \times \Bbb Z_6 \times \Bbb Z_9$ of index 3

I'm trying to solve the following problem from a past exam. Find the number of the subgroups of $P:=\Bbb Z_4 \times \Bbb Z_6 \times \Bbb Z_9$ of index 3. Here $\Bbb Z_m$ denotes $\Bbb Z/m\Bbb ...
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1answer
95 views

quotient Groups of different normal subgroups. [closed]

Let G have two normal sub groups N and M,|N|=|M|(so that |G/N|=|G/M|).Now consider their quotient groups G/N and G/M.Is it possible that for each g*n(g belongs to G and n belongs to N),there exists ...
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1answer
88 views

$U,H\leq G$ Subgroups. H of finite Index. Does $|U:H\cap U|\mid |G:H|$ hold?

Let $G$ be a group and $U,H\leq G$ subgroups, such that $H$ is of finite Index in $G$ (not necessary U, too). May $n=|G:H|$. One can easily show with an Injection between the two appropriate cosets ...
2
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1answer
120 views

Finite abelian groups in which quotients of same order are isomorphic

Let $G$ be a finite abelian group which is isomorphic to direct sum of some elementary abelian groups and a cyclic group such that all summands have coprime orders. Are quotients of its all subgroups ...
2
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3answers
123 views

$N$ is the only subgroup of order $|N|$

Let $G$ be a group of finite order. Let $N$ be a normal subgroup of $G$ and $\gcd(|N|,|G:N|)=1$. Show that $N$ is the only subgroup of order $|N|$. Here is my attempt at a solution: Suppose ...
5
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2answers
115 views

Presentation of $\mathbb Z_n\rtimes _{\phi}Q_8$

Regarding to this answer, I am looking for the presentation of $\mathbb Z_n\rtimes _{\phi}Q_8$, where $$Q_8=\langle a,b\mid a^4=1, a^2=b^2, ba=a^3b\rangle=\{1,a,a^2,a^3,b,ab,a^2b,a^3b\}, ...