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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2
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26 views

Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
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0answers
17 views

show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ .

suppose that $G$ is finite group and $p$ is a prime number,then show that $O^{p}(G)$ will generate with all $q$-sylow subgroups of $G$ where $q$ is arbitrary prime number and $q\neq p$ . ($O^{p}(G)$ ...
0
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0answers
25 views

I can't prove theorem 13.9 on finite permutation groups of Wielandt book.

I can't prove this theorem on finite permutation groups of Wielandt book. Theorem 13.9: Let $p$ be a prime and $G$ a primitive group of degree $n=p+k$ with $k\geq3$. if $G$ contains an element of ...
2
votes
1answer
44 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
0
votes
0answers
11 views

If $G$ has a $p$-complement $L$ then $L^{ G} = O^{P}(G)$.

Let $G$ be a finite group. If $G$ has a $p$-complement $L$ then $L^{ G} = O^{P}(G)$ (where $L^{ G}$ is the normal closure of $L$ in $G$ ). this question is from the book A course on group theory ...
0
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1answer
35 views

Finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ [on hold]

Give an example of a finite non-abelian group $G$ containing a subgroup $H_0\neq\{e\}$ with property that $H_0\leq H$ for all subgroups $H\neq\{e\}$ of $G$ This question is from Herstein Topics in ...
3
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0answers
43 views

Subgroups of $S_{n}$ of index $n$

We know that for $n\geq{5}$ any subgroup of $S_{n}$ of index $n$ is isomorphic to $S_{n-1}$. We know that by looking at the set of functions that fix a given $j$, we can obtain $n$ such subgroups ...
2
votes
2answers
34 views

Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)

I am a beginner in group theory and I'm looking for finite groups that satisfy some properties. The only example I've found so far is: $$G_{q,c} = \{ f: \mathbb{F}_{q} \to \mathbb{F}_{q}, z \mapsto ...
1
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3answers
97 views

Group of order $p^2$ is commutative with prime $p$

Please help me on this one: Let $p$ be a prime number, show that each group of order $p^2$ is commutative. If you do not mind at all, could you please not give me the elegant explanation, but ...
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0answers
16 views

prove that if $H$ is a $p^{'}$-sylow subgroup and $K$ is a $p$-group then $H^{G}=O^{p}(G)$.

suppose $G=HK$ which $H \leq H$ and $K\triangleleft G$ and $H \cap K =1 $ ,prove that if $H$ is a $p^{'}$-sylow subgroup and $K$ is a $p$-group then $H^{G}=O^{p}(G)$. these three last questions I ...
0
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0answers
24 views

prove that $O^{P^{'}}(G)=P^{G}=P[P,G]$.

suppose that $G$ is finite group and $p$ is prime number.prove that if $P$ is a $p$-sylow subgroup of $G$ then $O^{p^{'}}(G)=P^{G}=P[P,G]$ which $P^{G}$ is normal closure of $P$ in $G$ . any hint or ...
2
votes
0answers
29 views

Density of odd vs even group orders that are not forced to be simple by Sylow's Theorem

In Dummit & Foote's Abstract Algebra text, I've just solved the following two Exercises: Write a computer program which (i) gives each odd number $n<10,000$ that is not a power of a ...
1
vote
2answers
73 views

Number of subgroup of order $p^2$ in $\mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$

Let $G = \mathbb{Z}_{p^{2}} \times \mathbb{Z}_{p^{2}}$. How many subgroups does $G$ has of order $p^2$? I know there are only 2 cases of the subgroup H, H can be isomorphic to $Z_{p^{2}}$ or $ ...
5
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2answers
61 views

Normal subgroups in groups of odd order

I put the following question in my first-year algebra final this year: Suppose $G$ is a finite group of odd order and $N$ is a normal subgroup of order $5$. Show that $N\le Z(G)$. (By the way, this ...
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0answers
26 views

Isomorphism type of a finite matrix group

Give a known group to which $SO_5$ is isomorphic. Where $SO_5$ consists of $$\begin{pmatrix} \cos\frac{2k\pi}{5} & \sin\frac{2k\pi}{5} \\ -\sin\frac{2k\pi}{5} & \cos\frac{2k\pi}{5} ...
1
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1answer
36 views

Groups of Order 12 aren't Simple

Suppose $G$ is a group of order $12=2^2*3$. Let $n_p$ denote the number of Sylow p subgroups. Then $n_2$ is 1 or 3 and $n_3$ is 1 or 4. I want to show that one of them is one since if that is the case ...
6
votes
2answers
77 views

Galois group of the splitting field of the polynomial $x^5 - 2$ over $\mathbb Q$

We know that the splitting field of $x^5 - 2 $ over $\mathbb Q$ is $\mathbb Q(2^{1/5}, \rho)$, where $\rho$ is a fifth root of unity. Therefore, $\left[\mathbb Q(2^{1/5} , \rho) : \mathbb Q \right] = ...
1
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1answer
54 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
3
votes
1answer
47 views

Constructing an irreducible representation for a finite group

This is not a homework. Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like: ...
3
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0answers
59 views

Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
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0answers
31 views

If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
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2answers
55 views

A question in finite group theory with proof using representation theory

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises in this field, but I have a problem with an exercise: I want to prove ...
0
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0answers
66 views

The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
2
votes
1answer
39 views

A question in character theory of finite groups

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises from "A Course in the Theory of Groups by Robinson" and "character ...
0
votes
1answer
21 views

Normalizer and Centralizer coincide

I am working on the following question: Suppose $G$ is a finite group that has a cyclic 2-Sylow subgroup $H$. I want to show that the centralizer, $C_G(H)$, and $\text{normalizer,} \ N_G(H)$ coincide. ...
4
votes
1answer
51 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
4
votes
2answers
238 views

How can I test if an element g is a generator of a group G with a known number of elements, N?

Let's say $G$ has $1000$ elements. Without looping through each $g^m$, how can I show that $g$ is a generator? I've deduced that I must prove that the order of $g = N$, or in this case $1000$, but I'm ...
0
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0answers
19 views

To show that $S_4$ has a normal subgroup of order $4$ [duplicate]

WHow to show that $S_4$ has a normal subgroup of order $4$ ? . Please help .Thanks in advance
1
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1answer
64 views

Non abelian group of order 40

Let $G$ be a non abelian group of order 40 that is the direct product of two of its Sylow subgroups. Show that the center of $G$ has order 10. I tried 40=5×8 Since $G$ not abelian $G$ NOT EQUAL ...
0
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2answers
48 views

abstract algebra question with p-subgroup

Let $G$ be a finite group, $H<G$ a subgroup. Suppose that $H$ is a $p$-group, $p$ a prime, and $p\mid[G:H]$. Show that $p\mid[N_G(H):H]$ and $H<N_G(H)$. In particular, if $G$ is a $p$-group and ...
0
votes
1answer
34 views

Prove homomorphism and surjectivity of a function

I have a question about this exercise for my math study: Let $d, n \in\mathbb{Z}_{>0}$ with $d\mid n$. a) Prove that there is a homomorphism $g: (\mathbb{Z}/n\mathbb{Z})^*\rightarrow ...
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0answers
10 views

Is it ever possible for hypercomplexes to generate every element modulo a prime?

To start, we can take a well-chosen complex number, modulo a prime $p$, and generate every complex element modulo $p$. For example, if we take $(1+2i)^k \pmod 3$, each power of $k$ up to $(3 \cdot 3 ...
0
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1answer
28 views

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular.

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular. I proved that if $N$ is intransitive, then $G$ will be imprimitive and Frobenius but I don't ...
2
votes
1answer
59 views

Generalized Cauchy's theorem (group theory)?

I think there is a corollary for Cauchy's theorem such that a finite group $G$ contains a subgroup of order $n$ for each divisor $n$ of $|G|$ with a certain condition (I can't remember what it was). ...
0
votes
1answer
59 views

Number of Sylow subgroups

Prove that no group of order $56$ can be simple using steps ●finding sylows number 2-subgroups and sylow 7-subgroups ●explain why any of sylow 7-subgroups must intersect trivially, but this is not ...
2
votes
4answers
45 views

$G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$?

If $G$ is a finite abelian group and $m:=\max \{o(x):x \in G\}$ , then is it true that $o(x)|m , \forall x \in G$ ?
0
votes
1answer
45 views

Application of Chinese remainder theorem

from Chinese remainder theorem, we know that of $m,n \in \mathbf{Z}$, $(m,n) = 1$, then $Z_m \otimes Z_n \cong Z_{mm}$ as ring isomorphism, but how it related to the application that $\phi(nm) = ...
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0answers
39 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
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1answer
28 views

Evenness and oddness of group code weights

I'm doing exercises in Charles C. Pinter's book A Book of Abstract Algebra and I'm unable to solve problem 7 in section H of chapter 5 (subgroups). I think that there is a solution on this site but ...
0
votes
2answers
54 views

Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
3
votes
1answer
36 views

Finite abelian groups (application of structure theorem)

Problem Find all finite abelian groups that simultaneously have exactly $7$ elements of order $2$, exactly8 elements of order $3$, exactly $8$ elements of order $4$, at least an element of order ...
5
votes
1answer
47 views

Group of order $p^{n}$ has normal subgroups of order $p^{k}$

Q: Prove that a subgroup of order $p^n$ has a normal subgroup of order $p^{k}$ for all $0\leq k \leq n$. Attempt at a proof: We proceed by Induction. This is obviously true for $n=1, 2$. Suppose it ...
3
votes
3answers
57 views

A group action proof without group actions?

I am currently teaching an undergraduate abstract algebra course out of Saracino, Abstract Algebra: A First Course. Exercise 13.13 asks the following: Let $K$ be the subgroup $\{ e, (1, 2)(3, 4), ...
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0answers
30 views

Sylow Theorems for Symmetric (Permutation) Groups

The General Linear Group $GL(n,\mathbb{F}_p)$ has an interesting property that the proof of Sylow theorem for this group can be given which is based on the natural action of this group on the ...
1
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1answer
33 views

Non-existence of non-abelian simple groups of order $90$ by counting elements [duplicate]

I want to show that no group of order $90=2 \cdot3^2 \cdot5 $ is simple. Part (iii) of Sylow's Theorem gives $$n_2 \in\{1,3,5,9,15,45\} $$ $$n_3 \in \{1,10\} $$ $$n_5 \in \{1,6\} $$ I am aware of ...
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0answers
25 views

If $\sigma$ is a cycle of length $r$, then it has order $r$?

I should specify that $\sigma \in S_n$, the symmetric group. I've written down some permutations and it seems like their order correspond to their lengths. How can I prove this? I was thinking of ...
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0answers
27 views

Module of representation matrix

Can someone please show me why the module of any representation matrix in a one-dimensional representation of a finite group is equal to 1? and please define module of a representation as well. ...
0
votes
3answers
46 views

elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
0
votes
1answer
26 views

Find number of all $a \in G $ such that $o(a) =3$

Let $G$ be a group and $|G|= 51$ find number of all $a \in G$ such that $o(a)=3$ My solution : by this theorem : if $|G|=pq$ that $ p ,q$ are prime. If $ q\nmid p-1 $ then $\quad$ $G \cong \Bbb ...
0
votes
2answers
39 views

Group theoretical proof of $\varphi(rs)=\varphi(r)\varphi(s)$ through generators of the group.

Given a group $G=\langle a\rangle$ of order $rs$, with $(r,s)=1$, I showed there exist unique $b,c\in G$ such that $a=bc$ with $b$ of order $r$ and $c$ of order $s$. The latter is a direct consecuense ...