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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Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
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37 views

Does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$?

For $p$ prime, does $SL(2, \mathbb F_p)$ always have a subgroup of order $(p-1)(p+1)$? (It does seem to be the case for $p=3,5,7,11$) If yes, what are the generators? I guess this is simple, but my ...
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54 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
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34 views

Order of center of a p-group deduce abelian [duplicate]

Let $G$ be a group of order $p^n$, $p$ a prime. Suppose the center of G has order at least $p^{(n−1)}$. Prove that G is abelian. Attempt: use the class equation $|G|=|Z(G)|+ \sum_{i \in ...
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1answer
47 views

Embedding $S_n$ in $A_{2n}$

I want to embed the symmetric group $S_n$ into the bigger alternating group $A_{2n}$. How could I find such an injective homomorphism?
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1answer
48 views

Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
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2answers
104 views

Number of non-isomorphic groups of order $p^2$

The number of non-isomorphic groups of order $p^2$, where $p$ is a prime number is: 1. 1 2. $p$ 3. 2 4. $p^2$ What is simplest method to find number of non-isomorphic group? I read from various ...
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39 views

Composition series of nilpotent group

I found this problem in The Theory of Groups by Marshall Hall. Let the group $G$ be of order $p^rq^s$. If $G$ has two composition series $1 \unlhd A_1 \unlhd A_2 \unlhd \cdots \unlhd A_r \unlhd ...
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0answers
21 views

Direct product of groups and isomorphism [duplicate]

Let $A, B, C$ three groups such that $A \times C \cong B \times C$. I already know that if $A, B$ and $C$ are abelian and finite, then $A \cong B$. I think this result does not hold anymore if they ...
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2answers
37 views

Number of $n-1$-dimensional subspaces of $n$-dimensional space over finite field

I got a question with two parts. Let $V$ be a $n$-dimensional vector space over $\mathbb{F}_{p}$ - finite field with $p$ elements. a) How many $1$-dimensional subspaces $V$ has. b) How many ...
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1answer
39 views

Isomorphism of Quaternion group

Prove that $Q_{8} \cong H \rtimes G \Rightarrow H=\{e\}$ or $G=\{e\}$. My proof (is it correct?): We know that (it's a fact from the lecture): If $M \cong H\rtimes G$, then $H \cong K \unlhd M$ ...
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2answers
45 views

Isomorphism type of a finite group with respect to multiplication modulo 65

I'm the same guy revising for my group theory exam and posted a few days ago. I'm at the chapter on Finitely Generated Abelian Groups, and my prof gave this example which I don't quite understand: ...
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1answer
58 views

Exercise on Induced Representations of 1 dimensional complex representation

I'm having a hard time trying to solve the following problem coming from Eingof's book "Introduction to representation theory" (page 55 of the book in PDF format ...
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1answer
31 views

if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable.

if $G$ is a finite group and every sylow subgroup of it is cyclic,prove that $G$ is super-solvable. we have that $G$ is solvable,I want to show that all factors of derived series are cyclic. but no ...
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51 views

group without involution is 2-divisible

Let $G$ be an arbitrary torsion group without involutions. Show that $G$ is 2-divisible. I think it is enough to show $G$=$2G$ but i can't show why $2G$ can't be proper subgroups of $G$ ? Please ...
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62 views

Existence of a normal subgroup in G

Today on my algebra test I had such an exercise: Let $|G|=66$. Show that there is a normal subgroup in $G$ of order $3$. I am not even sure that's true. I wanted to show that $n_{3}$=1. But from ...
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20 views

Commutators inside center of factor subgroups

Let $G$ be a finite group. Assume that $K \subseteq L \trianglelefteq G$ with $K \trianglelefteq G$. Then $L /K \subseteq \textbf{Z}(G / K)$ if, and only if $[G,L] \subseteq K$. I know it is related ...
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1answer
74 views

Automorphism group of the Alternating Group - a proof

I was trying to read the following lemma which admit as an easy corollary the structure of the automorphism group of the alternating group on $n\geq 7$ elements. Anyway there are two points that ...
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1answer
26 views

Existence of integer $n > 2$ such that for any abelian group $G$ , $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ [closed]

Does there exist an integer $n > 2$ such that for any abelian group ( or at-least any finite abelian group ) $G$ , the set $G_n:=\{e\} \cup \{a \in G :o(a)=n \} $ is a subgroup of $G$ ?
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25 views

Prove that $Q_{8}'=\{1,-1\}$. Is my proof correct?

Prove that Prove that $Q_{8}'=\{1,-1\}$. My proof: $Q_{8}'\neq \{1\}$, because $Q_{8}$ is not abelian. $|Q_{8}/\{-1,1\}|=4$. So $Q_{8}/\{-1,1\} \cong \mathbb{Z}_4$ or $Q_{8}/\{-1,1\} \cong ...
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1answer
28 views

Showing that G is solvable

Let $|G|=200$. Show that G is solvable. My beginning of the proof: $|G|=200=2^3*5^2$ Let $n_5$ be the number of Sylow p-subgroups. Then $n_5|8$ and $n_5\equiv1 mod5$. And it implies that $n_5=1$. ...
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2answers
28 views

Injective Homomorphism from a group into $GL_n$

$|G|=n\ge 2<\infty,$ A group, I need to know which of the followings are true? $\exists$ allways an injective homomorphism from $G$ into $S_n$ $\exists$ allways an injective homomorphism from $G$ ...
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1answer
45 views

Trace group of a skew group algebra of a commutative domain

Let $R$ be a commutative noetherian domain that is also an algebra over a field $k$ Let $G$ is a finite group that acts on $R$ in a non-trivial way. Let $A=R*G$ be the skew group algebra of this ...
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29 views

Generating a group by its $q$-elements.

Let $G=PQ$ be a solvable group where $P$ and $Q$ are $p$-subgroup and $q$-subgroup of $G$ respectively. Also suppose that $Q$ is not normal in $G$. Is it true that the group generated by all ...
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56 views

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy's theorem I can show that there are elements of order $2$ and $3$ but cant proceed ...
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34 views

Intersection theorems for a certain type of subsets of integers modulo $N$

I've been working on something with integers modulo $N$ and have sort of hit a roadblock where I'd like to have some references. The particular problem goes as follows. We have a system $\mathcal{S}$ ...
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1answer
33 views

Existing of such automorphism subgroup

Let $G$ be abelian group and $Aut(G)$ be the automorphism group of $G$ , I am looking for a nontrivial subgroup $H$ of $Aut(G)$ such that $gcd(|H|,|G|)=1$. Does such subgroups always exit ?
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1answer
18 views

semi-direct product of subgroups of $D_6$

I have a hexagon with edges $A,B,C,D,E,F$ and its symmetry group $D_6$. I want to prove that $D_6 = H \rtimes M$ given the subgroups $H = \{ g \in D_6 \,\,|\,\, g \text{ permutes } \{A,C,E\} \}$ and ...
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1answer
25 views

$D_6$ and cycle notation problem

I have a hexagon with edges $A,B,C,D,E,F$ and I want to work with its symmetry group $D_6$ in cycle notation. My calculations don't yield consistent results. For example, I correctly get $r^4 \cdot ...
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24 views

Can a non-abelian group of order $105$ have trivial center ?

Can a non-abelian group of order $105$ have trivial center ?
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1answer
95 views

Group and Ring Homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$

Let $p$ be prime with $p > 2$. (a) Determine the number of group homomorphisms between $\mathbb{Z}_p$ and $\mathbb{Z}_{2p}$ (b) Determine the number of ring homomorphisms between $\mathbb{Z}_p$ ...
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1answer
51 views

Why is $D_5$ a subgroup of the icosahedral group

According to Wikipedia $D_5$ is a subgroup of the group of rotational symmetries of an icosahedron: http://en.wikipedia.org/wiki/Icosahedral_symmetry. I know this isn't very rigorous, but intuitively ...
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1answer
37 views

Symmetric polynomials, group of permutations

Could somebody give me a clue, related to the possible solution of the problem? Let's denote a polynomial $f(x_{1}, x_{2}, x_{3}, x_{4}, x_{5})=x_{1} x_{2} x_{3} +x_{2} x_{3} x_{4} + x_{3} x_{4} ...
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1answer
23 views

Subgroup lattice of $U(12)$

$U(12)$ is not cyclic. Order of $U(12)$ is $4$. By Lagrange's Theorem, order of a subgroup must divide the order of the group. Hence any subgroup of $U(12)$ must have order $1, 2 \text{ or } 4$. ...
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2answers
40 views

Order of $A \in GL(n,\mathbb Z_p)$ cannot exceed $p^n-1$ ?

If $A \in GL(n,\mathbb Z_p)$ then is it true that order of $A$ cannot exceed $p^n-1$ ?
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1answer
139 views

How do I construct a nonabelian group of order 1575?

I think that it should be a semidirect product of the direct product of any two of the three groups $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/25\mathbb Z$ and the other one. But ...
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45 views

Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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1answer
162 views

Cancellation of Direct Product in Grp

I'm thinking to the famous problem of cancellation property in Grp, i.e: $$G_1 \times G_2 \cong G_1 \times G_3 \Rightarrow G_2 \cong G_3. $$ Clearly there are many counterexamples like $\prod_{i \in ...
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1answer
61 views

Surjections from free groups

I am stuck on the following: How do I go about finding surjections from the free group of rank 2 $\mathbb{F}_2 = \mathbb{ Z∗Z}$, to the finite group of two elements $\mathbb{Z}_2$. Also, how would ...
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30 views

infinite version of lagrange theorem [duplicate]

Let G be a group and H and K subgroups of G. If K ⊂ H ⊂ G and K has finite index in G, then prove [G : K] = [G : H][H : K]. Obviously if we know G is finite, then we are done by Lagrange Theorem. ...
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1answer
34 views

example for permutizer group

permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H. You can help us give an example?
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26 views

if group $G$ has three elements $a$,$b$,$c$ with the property that the order of each two of them is coprime,and $c=ab$ then $G$ is not solvable.

if group $G$ has three elements $a$,$b$,$c$ with the property that the order of each two of them is coprime,and $c=ab$ then $G$ is not solvable. this question is about my last question posted which ...
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1answer
51 views

Prove that every group $G$ whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$ is not solvable [closed]

Prove that every group $G$, whose order is the form $|G|=p_1^{\alpha_1}p_2^{\alpha_2}p_3^{\alpha_3}$, where $p,q,r$ are distinct prime numbers and $\alpha_i >1$, is not solvable. Any hint or ...
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0answers
101 views

Permutation group of a set

If we let $G$ be a finite permutation group of a finite set $X$ and assume that $G$ has exactly 2 orbits of the same cardinality, how can we show that there is some permutation in G that has no cycles ...
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1answer
33 views

Composition factors of linear groups

The following problem comes from an algebra exercise and since two days or so, I am not able to find a satisfying solution: Let $p$ be a prime with $p \geq 5$. Let $F_p$ denote the field with $p$ ...
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1answer
43 views

Proving a subgroup is normal

Problem Let $G$ be a group with $|G|=pm$, $p$ prime and $p \geq m$. Suppose there is $H$ subgroup of $G$ with $[G:H]=p$. Show that $H$ is normal. This problem was given to me in class just after ...
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20 views

Length function

Let $W$ be a Coxeter group with simple system $S$, positive system $P$ and root system $R$. Then $S\subset P\subset R$. Let $\lambda:R\rightarrow\{0,1\}$ be the characteristiv function of $P$, in ...
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30 views

If all sylow subgroups are cyclic, prove that G is solvable

I came across a statement which I am unable to prove by myself that if $G$ is a finite group then if all its sylow subgroups are cyclic, prove that G is solvable. If it has been asked before please ...
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36 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
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1answer
17 views

Show that the following group action has a non-zero singleton orbit.

Let F be a finite field of characteristics prime p.Let G be a group of order $p^r$ for some r.Let G acting on $F^n$ for n>1.Then show that there exist a non-zero vector in $F^n$ whose orbit will be ...