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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2answers
93 views

How do I know there are only 5 different groups of order 8? [duplicate]

How many different groups are there in order 8? And how do I know which groups they are? I mean, is there anyone can teach me to calculate them? I want a proof, thank you! They are $C_8$, $D_4$, ...
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7answers
3k views

Are two finite groups of the same order always isomorphic?

Are two finite groups of the same order always isomorphic? Some simple example would be great!
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0answers
132 views

Two questions in Isaacs' book Finite Group Theory

I am reading Isaacs' book finte group theory, and I have two questions. in page 90, there is a Wielandt's theorem (if $G$ has a nilpotent Hall $\pi$-subgroup, then all Hall $\pi$-subgroups of $G$ ...
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1answer
133 views

Let $A$ be an abelian group. $A(p)$ is a p-group if $A(p)$ is finite.

Let $p$ be a prime number, let $A$ be an abelian group, define $A(p)$ to be the subgroup of all elements that have power of $p$ order, i.e. $x \in A(p) \iff x\in A \ \wedge \ p^ex = 0$, in additive ...
0
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1answer
128 views

Finitely generated abelian group isomorphic to infinite abelian group?

Lang's Algebra says that if an abelian group $A$ is free and finitely generated by $(x_i), i=1,\dots, n$ , then it is isomorphic to $\mathbb{Z}x_1 \bigoplus \cdots \bigoplus \mathbb{Z}x_n$, which is ...
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1answer
65 views

Two elements are in the same coset of $S$ iff their difference is in $S$

Assume $S$ is a subgroup of group $G$ How to prove this: Two elements are in the same coset of $S$ iff their difference is in $S$
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1answer
42 views

how do I find all Elements of a Group?

I am given a Group $\mathbb{Z_{11}^*}$. a multiplicative group. How do i find all elements of this Group?
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3answers
250 views

For any finite group, there is a homomorphism whose image is simple

This is for homework. The question asks "Show that, for any finite group $G$, there is a homomorphism $f$ such that $f(G)$ is simple." My thought was this. Since $G$ is finite, there are only a ...
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1answer
212 views

Prove that a group is a quaternion group.

The representation of the Quaternion group is $$\mathrm{Q} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle.$$ Does this imply that as long as I have found a group with $4$ ...
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3answers
90 views

Exercise: product of transposition

How would I go about computing $$(1 2 3)\cdot(12)(34)$$ I know the definitions but I do not know how to apply them here. This is rather strange and odd-looking to me. I know I have to construct a ...
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1answer
217 views

Solvable group of certain order

‎We ‎know ‎that ‎in a non-abelian ‎group ‎of ‎order ‎‎$p^2q‎$ ($‎‎p$ ‎and ‎$q‎‎$‎ ‎are distinct primes)‎‎, ‎ if ‎$p>q‎‎$ and it's Sylow $p$-subgroup is elementary abelian $p$-group‎‎, ‎then ‎one ...
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1answer
78 views

what is multiplicative group of all integers coprime with $N$ called?

what is multiplicative group of all integers coprime with $N$ called? I am not sure the tags are correct or not!
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2answers
82 views

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.
2
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1answer
57 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
122 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
823 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
124 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
345 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 ...
2
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2answers
81 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 ...
6
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1answer
170 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
92 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
132 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
129 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 ...
2
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1answer
100 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
244 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 ...
3
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1answer
173 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 ...
0
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3answers
51 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
67 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
80 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 $g^k$ ...
4
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2answers
104 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
142 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
435 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
65 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
49 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?
4
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1answer
75 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 ...
4
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4answers
743 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 ...
6
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1answer
72 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 ...
2
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1answer
95 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
200 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
50 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
202 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
197 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 ...
0
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1answer
43 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 $b$.
2
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1answer
104 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!
3
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2answers
97 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
69 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 ...
4
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3answers
235 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 $m$ ...
0
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2answers
246 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
79 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?
1
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1answer
119 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 $D_{2n}$?