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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574 views

Proving that in an infinite cyclic group order of every element is infinite

Prove that in an infinite cyclic group order of every element ($\ne e$) is infinite. I have tried proving this way : If there exists an element of finite order, then it must generate a finite ...
2
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1answer
410 views

Show that the intersection of any two distinct Sylow $2$-subgroups of $G$ has order $8$

Suppose that $G$ is a group of order $48.$ Show that the intersection of any two distinct Sylow $2$-subgroups of G has order $8.$ Let $H,K$ be two distinct Sylow $2$-subgroups of $G.$ Then ...
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2answers
441 views

Does every finite group has finite cyclic subgroup?

Does every finite group of size at least 2 has finite cyclic subgroup of size at least 2? Lagrange's theorem doesn't help, because it doesn't say anything about existence of subgroup of some ...
4
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1answer
179 views

Intuition behind Upper central series of Group

To construct abelian groups from non-abelian groups people have introduced the notion of Commutator subgroup and solvability of groups. Can somebody explain what is the intuition behind in ...
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3answers
199 views

On simple groups $G$, where $2\mid |G|$, $4\not\mid |G|$

The (old) exam I'm looking at has the following problem: Suppose the order of $G$ is even, but is not divisible by $4$. Prove that $G$ is not simple. A group with $2$ elements is clearly a ...
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2answers
266 views

Finite Group generated by the union of its Sylow $p_i$-subgroups

Let $\lbrace P_i : i\in I \rbrace $ be a set of Sylow subgroup of a finite group G, one for each prime divisor $p_i$ of $|G|$. Show that $G$ is generated by $\bigcup P_i$ from Rotman "An ...
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1answer
447 views

Is a finitely generated torsion group finite in general?

This is not a duplicate of Can a finitely generated group have infinitely many torsion elements? There he asks specifically about FC-groups. Is a finitely generated torsion group finite in general? ...
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2answers
197 views

Finding the kernel of an action on conjugate subgroups

I'm trying to solve the following problem: Let $G$ be a group of order 12. Assume the 3-Sylow subgroups of $G$ are not normal. Prove that $G\cong A_4$. Here's my attempt: let $\mathscr S$ be ...
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1answer
58 views

Decide existence of finite non-abelian group $G$.

Decide existence of finite non-abelian group $G$ such that: $ ( \exists a,b \in G$) and $\mbox{ord}(a) = \mbox{ord}(b) = 2$ for above $a,b$: $\mbox{ord}(ab) > 2$ I suppose that correct example ...
8
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1answer
290 views

finite subgroups of SO(3)

As is well-known, all finite subgroups of $SO(3)$, except for cyclic and dihedral groups, are isomorphic to $A_4$, $S_4$, or $S_5$. The classical proof of this fact uses the geometry of regular ...
2
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1answer
115 views

Which of the following groups are isomorphic?

Which of the following groups are isomorphic? $\mathbb Z_{24}, \mathbb Z_{4}\times \mathbb Z_{6}, S_4, A_4\times\mathbb Z_{2}, \mathbb Z_{8}\times \mathbb Z_{3}, D_{12},D_6\times \mathbb Z_{2}$ ...
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2answers
780 views

How every element of a group generates a cyclic subgroup?

It is given that every element of a group generates a cyclic subgroup. I am not able to see how. If, let's say, $H=\langle a \rangle, a\in G$ then, $H={\{...,-a^{-2},-a^{-1},a^0,a^1,a^2,...\}}$. Then ...
4
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0answers
126 views

$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse. [duplicate]

If $G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse. Then we have show that $$T_x=x^{-1}$$ for all $ x \in G$ and $G$ ...
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3answers
2k views

Does every group whose order is a power of a prime p contain an element of order p?

I need to know if every group whose order is a power of a prime $p$ contains an element of order $p$? Should I proceed by picking an element $g$ of the group and proving that there is an element in ...
5
votes
1answer
318 views

Subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$

I am trying to show that the subgroups of $S_n$ of index $n$ are isomorphic to $ S_{n-1}$, if $n\ge 2$. I tried to do this as follows: Let $G < S_n$ be a subgroup of index $n$, and let it act on ...
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1answer
72 views

Understanding a proof about finite $p$-groups

I can't follow the reasoning of the author,in this proof: let $G$ be a finite $p-$group. If $H$ is a proper subgroup of G, then $H<N_G(H)$ (clearly $N_G(H)$ is the normalizer of $H$ and p is ...
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2answers
1k views

Subgroups of a direct product

Until recently, I believed that a subgroup of a direct product was the direct product of subgroups. Obviously, there exists a trivial counterexample to this statement. I have a question regarding ...
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3answers
108 views

nonisomorphic groups whose quotients are isomorphic

Assume that we have two groups $A$ and $B$ such that $C \subset A$ and $C \subset B$ where $C$ is a normal subgroup of both $A$ and $B$. If we have that $A/C \cong B/C$ is it true that $A \cong B$? I ...
2
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1answer
78 views

filling in the gaps of this proof involving the fundamental theorem of finitely generated abelian groups [duplicate]

I have an abstract algebra proof I can not complete. The proposition goes as follows: Consider a group $G$ of order $m$. If $n$ divides $m$, prove that $G$ has a subgroup of order $n$. This question ...
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1answer
86 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
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1answer
55 views

$G=C_{p}\times C_{p^2}$, describe $\mathrm{End}(G)$

I was asked to describe the group of the endomorphism of $G=C_{p} \times C_{p^2}$, with p prime ($C_n$ is the cyclic group of order $n$). I started setting (g,1) and (1,h) as generators of the ...
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1answer
239 views

Dihedral group as a direct product

In a problem from a past exam I am asked "When can $D_n = \langle r,s\mid r^n = s^2 = (rs)^2 = 1\rangle$, the dihedral group of order $2n$, be expressed as a direct product $G\times H$ of two ...
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1answer
3k views

Normal subgroups of dihedral groups

In relation to my previous question, I am curious about what exactly are the normal subgroups of a dihedral group $D_n$ of order $2n$. It is easy to see that cyclic subgroups of $D_n$ is normal. But ...
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0answers
74 views

(some) Dihedral groups have a right transversal isomorphic to some dihedral group.

Let $D_{2n}=\langle a,b;a^n,b^2,(ba)^2\rangle$ denote the Dihedral Group of order $2n$. $H=\langle a^m,b\rangle$ be a subgroup of $D_{2n}$ of even index $m$ such that $m$ divides $n$. Choose a right ...
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95 views

character of induced representation and the lifting representation

$A:H\rightarrow GL_n(\mathbb C)$ is a representation. $H$ is a subgroup of $G$. We define the lifting $\overline A(\sigma)=A(\sigma)$ if $\sigma\in H$, $\overline A(\sigma)=0$ if $\sigma\in G-H$. I ...
2
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1answer
206 views

Geometric interpretation of the dihedral subgroups of a dihedral group

I learned that a subgroup of $D_n = \langle r,s \mid r^n=s^2=(rs)^2=1 \rangle$, the dihedral group of order $2n$, is either cyclic or dihedral itself, and that a subgroup of the latter kind is of the ...
2
votes
2answers
421 views

What is the size of the normalizer of a subgroup generated by a $p$-cycle in a symmetric group?

Question: What is the size of the normalizer of a $p$-cycle (prime $p$) in the symmetric group $S_n$ ($n \geq p$)? If $n<2p$, we can actually find the size $N:=|N_{S_n}(\langle (12\cdots p) ...
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0answers
65 views

Is a certain subgroup $S\leq G$ in the center of $G$, $S\leq Z(G)$?

All groups considered are finite. Let $A$ be a group such that $A=A'\left<x\right>$, where $A'$ is the commutator group and $\left<x\right>$ is cyclic of order $p\in\mathbb{P}$. How can I ...
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1answer
209 views

Are the Groups having all subgroups of same order conjugate classified?

As we know, Alternating Groups $A_n$ of degree $n\leq 5$, Dihedral Groups $D_{2n}$ of order $2n$ (for odd natural number $n$) and Cyclic Groups $C_n$ of order $n$ (for positive integer $n$) have ...
4
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1answer
153 views

Number of Sylow $p$-subgroups is congruent to $1$ modulo $p^a$.

Let $G$ be a finite group having more than one Sylow $p$-subgroup. Let $p^a$ be the least element of $\{|P|/|P\cap Q|: P,Q\in \operatorname{Syl}_p(G),P\neq Q\}$. Prove $n_p\equiv 1\pmod {p^a}$. ...
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2answers
285 views

Generators of Sylow $2$-subgroups of $ S_8 $

I know that $ S_8 $ has Sylow $2$-subgroups of order $ 2^7 $. I have no idea about their generators.
6
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2answers
175 views

Problem about finite group and conjugation

Let $G$ be a finite group with order $n\geq 7$ and H be a proper subgroup of $G$ then prove that there exists at least $3$ elements of $G$ that do not conjugate to any elements of $H$ Any idea? I ...
2
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3answers
148 views

$G$ finite abelian, $\exists H < G : |G/H|$ is prime?

Let $G$ be a finite abelian group. Is there a subgroup $H < G $ s.t. the quotient $ G/H$ has prime order?
5
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2answers
96 views

There is no core free subgroup of order $p^2$ in a group of order $p^4$

By the classification of group of order $p^4$ ($p$ odd prime) from Burnside's book it seems to me that there is no core free subgroup of order $p^2$ in a group of order $p^4$. If I am not wrong there ...
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1answer
92 views

Proving that doesn't exist subgroup H with order 6 in $A_4$ [duplicate]

Let $G=A_4$. Prove that does not exist subgroup $H\le G$ s.t $|H|=6$. I don't know from where to start (maybe I need to prove that if so $H\triangleleft A_4 $?) Any hint will be appreciated.
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2answers
62 views

Want to clarify whether I am correct or not, $\Phi(G) \subseteq \Phi(H)$?

I Want you to clarify whether I am correct or not regarding following question. I will be thankful to you for telling me if I am wrong: Let $G$ be finite group and $\Phi(G)$ denotes its frattini ...
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1answer
178 views

Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Find all numbers $n$ such that $S_7$ contains an element of order $n.$ Identity is the only element of order $1.$So $n=1$ is possible. Case 1: Elements that can be written as a unique cycle of ...
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1answer
56 views

how $a+_m(b+_mc)=a+_m(b+c)$?

I am trying to show that the set of first m non-negative integers is a group under the composition of addition modulo $m$. I need some help understanding this step - $$a+_m(b+_mc)=a+_m(b+c)$$ It ...
5
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1answer
536 views

Two elements that generate a group of order $22$

A group $G$ of order $22$ contains elements $x$ and $y$, where $x \neq 1$ and $y$ is not a power of $x$. Show that the subgroup generated by these two elements is the whole group. I'm not ...
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0answers
197 views

Cayley theorem to prove that a group of order $2^mk$, $k$ odd, can't be simple

I have to solve this exercise WITHOUT Sylow theorems and Cauchy Lemma. In fact this exercise is given in the Cayley's theorem section. let $G$ be a group of order $2^mk$, where $k$ is odd. Prove ...
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4answers
101 views

An innocent homomorphism between groups

There does not exist a nontrivial homomorphism $h:(\mathbb{Z}_8,+)\rightarrow(\mathbb{Z}_3,+)$. I am trying to understand this well-known fact. Let $h(x) = x$. Then certainly $h(a+b) = h(a) + ...
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1answer
43 views

The relationship between prime divisor of the number of cyclic subgroups of order $2$ with prime divisor of a finite group

Let $G$ be a finite non-abelian group and $k$ be the number of cyclic subgroups of order $2$. Why is every prime divisor of $k$ a prime divisor of the order of $G$?
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3answers
123 views

Does representation theory exists without Groups?

I need to know: is representation theory all about Groups? Is it necessary to be a finite group? Does representation theory exists without Groups? For example is there sample where representation is ...
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1answer
46 views

Automorphism group of groups of order 42

I would like to find an abstract solution for the following question: If $G$ is a group of order $42$, then $5$ does not divide the order of ${\rm Aut}(G)$.
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2answers
150 views

Computing bicharacters of (small) finite groups

I'm trying to find some finite groups with certain properites (hopefully of small order; no more than 100, I suspect), and one of the things I need to look at are all of its bilinear bicharacters: ...
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2answers
150 views

Number of conjugates of $x \in G$ in $H \triangleleft G$

Let $G$ be a finite group, let $H$ be a normal subgroup of prime index, and let $x \in H$ satisfy $C_H(x) < C_G(x)$. If $y \in H$ is conjugate to $x$ in $G$, then $y$ is conjugate to $x$ in $H$. ...
3
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1answer
158 views

An automorphism of order 2 which fixes only the identity

Let $G$ be a group. Assume there is an element $\phi\in\text{Aut}(G)$ such that $\phi(x)=x\Rightarrow x=e$, where $e$ is the identity of $G$, and that $\phi^2$ is the identity automorphism on $G$. I ...
6
votes
1answer
168 views

Simple subgroup of Symmetric Group

I have the following question: Let $n\geq5$, and suppose that $G$ is a simple subgroup of $S_{n+1}$ of index $k$. Show that if $k\leq2n+2$, then $G=A_{n+1}$ or $G$ is isomorphic to $A_n$. I have ...
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0answers
54 views

Condition for the number of distinct solutions over GF($q$)

Assume that we have $p$ sets $\left\{ {{m_i}} \right\}_{i = 1}^p$ with given cardinalities $\left\{ {{K_i}} \right\}_{i = 1}^p$, $1 \le {K_i} \le q$, where $q$ is a power of $2$. What I'm trying to do ...
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0answers
68 views

Calculating all minimal generators of a finite group

Is there a way to calculate ALL minimal generators of a given finite group? (A generating set of a group is termed minimal or irredundant if any proper subset of the generating set, generates a ...