Tagged Questions
0
votes
1answer
62 views
1
vote
2answers
75 views
$H$ must contain every Sylow $p$-subgroup of $G$
G is finite and has a prime factory. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that
4
votes
2answers
60 views
If the order of a finite abelian group is not divisible by a square, show that the group must be cyclic.
If the order of a finite abelian group is square free, show that the group is cyclic.
This is a question from "basic abstract algebra" by bhattacharya
3
votes
3answers
77 views
Sum of two squares in a $\Bbb Z/p\Bbb Z$
I need to show that every element in $\Bbb Z/p\Bbb Z$ can be written as a sum of two squares. The case $p=2$ is trivial and $0$ is always $0^2 + 0^2$. So all I have to do is show that every element of ...
3
votes
1answer
40 views
order of $\langle (123) , (234) \rangle$
As homework the teacher asks us to determine how many elements are there in $\langle (123) , (234) \rangle \subset S_4$ .
I've started doing all the multiplications between the elements, and I've ...
4
votes
1answer
57 views
Action of $S_4$ in $S_4/S_3$
Let $G = S_4$, $H = S_3$, $X = G/H$ be the set of right cosets of $H$, $x = (14)H$ and $G $ acts on $X$ by conjugation. Compute $\mathscr{O} (x)$ and $G_x$ (the stabilizer of $x$).
I've got a ...
1
vote
5answers
66 views
Show that If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$
If $|G|=p^2$ and $H\leq G$ with $|H|=p$, for $p$ any prime, then $H$ is normal in $G$.
I am sort of stuck with this proof and I would appreciate a hint (not a full solution, please!). Preferably, ...
2
votes
2answers
76 views
Finitely generated abelian group.
Let $G,H,K$ be finitely generated abelian groups. If $G \times K$ is isomorphic to $H \times K$, then $G$ is isomorphic to $H$.
What I have thought is that fundamental theorem of abelian groups can ...
7
votes
2answers
138 views
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
Let $G$ be a finite group. For each positive integer $m$, if $x^{m}=e$ has at most $m$ solutions in $G$, $G$ is cyclic.
What I have thought is that $n=\sum_{d\mid n}\phi(d)$ can be used to solve ...
2
votes
2answers
66 views
Order of kernel of a homomorphism.
Let $C_n$ denotes the cyclic group of order $n$ and let $\phi:C_{52}\rightarrow C_{52}$ be the homomorphism $\phi(x)=x^7$. What is the order of kernel of $\phi$?
I know that $ker\phi=\left\{x/ ...
3
votes
1answer
64 views
Exercise 2.15 M.Isaacs' Character theory of finite groups
I'm beggining to study character theory, and i'm doing some problems from Isaacs' Character theory book.
I would need some help with this one:
(2.15): Let $\chi\in \operatorname{Irr}(G)$ be ...
1
vote
1answer
69 views
Exercise 2.8 M.Isaacs' Character theory of finite groups
I'm a starter at character theory. I'm trying to do this exercise:
(2.8) Let $\chi$ be a faithful character of a group $G$. Show that $H\subseteq G $ is abelian if and only if every irreducible ...
0
votes
1answer
57 views
Question about the fundamental group of simplicial complex and the universal cover.
Can any one give me any idea about how to solve this problem?
Suppose we have a simplicial complex G which is finite connected.
(1)The fundamental group of G is finite;
(2)The universal cover of G is ...
2
votes
1answer
27 views
Maps of representations
Let $G$ be a group and $V_j$ , $j =1,2$ be irreducible representations of $G$. Show that any map $\phi: V_1 \rightarrow V_2$ of representations is either an isomorphism or zero.
The Hint I got was: ...
4
votes
1answer
74 views
How to prove one-one and onto?
Let $G$ be a group of odd order. Show that the function $\phi:G \to G $ given by $\phi(g)=g^2$ is one-one and onto.
To prove one-one, I did
$\phi(g_1)=\phi(g_2)$ implies $g_1^2=g_2^2$. Somehow I got ...
0
votes
1answer
39 views
Construct a complete system of representatives for the left cosets of $H_2$ in $G$.
Let $G$ be a finite group. Let $H_2\subseteq H_1$ be subgroups.
Let $R$ be a complete system of representatives for the left cosets of $H_1$ in $G$.
Let $S$ be a complete system of representatives ...
4
votes
2answers
55 views
Is it true to say $Z(G)\subseteq N_G(H)$?
Let $p$ be an prime number and $G$ a group of order $p^n$ and $H$ be subgroup of $G$ of order $p^{n-2}$ and which is not normal in $G$. Is it true to say that $Z(G)\subseteq N_G(H)$?
3
votes
3answers
41 views
$D_6$ as permutation group
I am trying to solve some exercises for a course in representation theory. We are studying finite groups and I have an exercise about the dihedral groups $D_n = ...
2
votes
1answer
83 views
Completing a Cayley table with few given spaces
\begin{array}{ccc}
* & \textbf{1} & \textbf{2} & \textbf{3} & \textbf{4} & \textbf{5} & \textbf{6} \\
\textbf{1} & 1 & & & & & \\
\textbf{2} & ...
2
votes
4answers
123 views
Finite Group Proving finite order of elements and Subgroup Question
The question is as follows
Let G be a finite group.
(i) Prove that every element of G has finite order.
For this want to use the idea that if G is finite then for a in G, $a^{n}$ = $e$ for some n ...
1
vote
2answers
188 views
How many non isomorphic groups of order 30 are there?
Let $|G|=30$. I have prove that there is the only subgroup of order $15$, which I'll denote $H$. Now I do know how to classify the group. After thinking, I made the following steps.
1) Possible ...
0
votes
1answer
88 views
Why must a finite symmetry group be discrete?
I'm having trouble justifying why a finite symmetry group is discrete. Can someone help?
2
votes
1answer
80 views
Questions about products of $p$-cycles.
Let $p$ be a prime and let $n$ be an integer such that $n \le p$.
a) Let $\alpha$ be a p-cycle in $S_n$. Prove that $\alpha^j$ is a p-cycle for $1 \le j \le p − 1$.
b) Assume that $2p \le ...
4
votes
2answers
57 views
Equivalence Relation in a Subgroup
Let $G$ be a finite group and $H$ a subgroup. Define a relation on $G$ by $$a\sim b\iff b^{-1}a \in H.$$
(0.) Show that this is an equivalence relation.
(i) Prove that for this relation $$[a] = ...
-3
votes
1answer
128 views
A multiple choice question on finite group
Let $G$ be a finite group such that $Z(G)=1$. Let there exist $m$ such that $G$ has a unique element of order $m$. Which of the following statements is true?
(a) $m=1$
(b) $m$ is prime
(c) ...
2
votes
5answers
364 views
Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.
(i) Show that $S_4/V$ is isomorphic to $S_3 $, where $V$ is the Klein Four Group.
I understand isomorphism to be a bijective homomorphism but I'm unsure how one would go about proving this. $S_4/V$ ...
9
votes
2answers
127 views
The existence of a group automorphism with some properties implies commutativity.
Let $G $ be a finite group, $T$ be an automorphisom of $ G $ st $ Tx = x \iff x=e $. Suppose further that $ T^2 =I $. Prove that $ G $ is abelian.
I was thinking if I show $ T aba^{-1} b^ ...
2
votes
4answers
147 views
Let G be finite group, which has an even number of elements.
Let $G$ be finite group, which has an even number of elements. Show that at least for two (distinct) elements $g,h$ of group $G$ one has $g*g = e$ and $h*h = e$.
I just started learning algebra and I ...
1
vote
3answers
70 views
On the center of a finite group $G$ with a normal Sylow subgroup
Suppose nonsolvable finite group $G$ has a normal Sylow $p$-subgroup. I would like to know whether center of the group $G$ is nontrivial?
1
vote
5answers
177 views
Expression as a product of disjoint cycles
Let $\alpha = (9312)(496)(37215) \in S_n, n \ge 9$. Express $\alpha$ as a product of disjoint cycles.
I know this is probably a really easy question, but my professor didn't elaborate on how to ...
0
votes
0answers
41 views
A question on a group of order 28 [duplicate]
Possible Duplicate:
A group of order 28 with a normal Sylow 2-subgroup is abelian
Let $G$ be a non abelian group of order 28. Is it true that $G$ contains a normal subgroup of order 4?
1
vote
1answer
56 views
Burnside's Action Equivalence Criterion
Theorem: Actions of a finite group G on finite sets $X$ and $Y$ are equivalent if and only if $|X^H|=|Y^H|$ for each subgroup $H$ of $G$
I've proved the finite case inductively but I'm wondering ...
3
votes
1answer
114 views
On Group action and blocks of subgroups of the symmetric Group
this exercise is from Dummit and foote , page 117 , # 7.d
prove : a transitive group $G$ is primitive on $A$ iff for each $a \in A$ , the only subgroups of $G$ containing $G_a$ are $G_a$ and $G$
...
3
votes
1answer
66 views
Chief series of a group
I need help in checking some reasoning in an answer.
Let $G$ be a group with order 180. Suppose G is a group with chief series $G = G_0 \geq G_1 \geq G_2 \geq \cdots \geq G_r = \{1\}.$ What are the ...
2
votes
2answers
85 views
Exponent of a direct product of cyclic groups
I have an answer to a homework question that I am not sure is correct. The question is show that if $G \cong C_{n_1} \times C_{n_2} \times \cdots \times C_{n_k}$ for positive integers $n_1, n_2, ...
0
votes
2answers
57 views
Principal ideal ring if and only if it is both a Noetherian ring and a Bezout ring.
How can I prove it?
Principal ideal ring if and only if it is both a Noetherian ring and a Bezout ring.
I think that The fact that a Noetherian Bezout ring is a principal ideal ring follows by a ...
3
votes
2answers
152 views
Non-Abelian simple group of order $120$
Let's assume that there exists simple non-Abelian group $G$ of order $120$. How can I show that $G$ is isomorphic to some subgroup of $A_6$?
1
vote
2answers
55 views
Given $\tau\in{S_n} $ a cycle of length $m$ that $\tau^{m} = e$
I'm trying to rigorously prove that given a cycle $\tau\in{S_n} $ of length $m$, then $\tau^{m} = e$ where $e$ is identity.
The funny thing is that I know why it works and understand it intuitively ...
3
votes
2answers
143 views
A normal subgroup $H$ with $[G:H]$ coprime to $p$ contains every Sylow $p$-subgroup of $G$.
I was working on the following question:
Let $p$ be a prime factor of the order of a finite group $G$. If $H$ is a normal subgroup $G$ whose index is not a multiple of $p$, show that H must ...
0
votes
0answers
22 views
how to calculate the orbit sizes of $A_{4}$ in $PSL(2,q)$
let $q$ be a prime power $q\equiv 3\pmod 4 $and let $X=GF(q)\cup\lbrace\infty\rbrace$.Then the set of all mappings
$g:x\rightarrow {ax+b \over cx+d}$
on $X$ such that $a,b,c,d\in GF(q),ad-bc$ is a ...
3
votes
1answer
124 views
How to show that the 3-cycles $(2n-1,2n,2n+1)$ generate the alternating group $A_{2n+1}$.
I'm asked to show that the 3-cycles $(1,2,3),(3,4,5),(5,6,7),...$ and $(2n-1,2n,2n+1)$ generate the alternating group $A_{2n+1}$.
I know the 3-cycles produce the group $A_n$, and it seems like I have ...
4
votes
1answer
124 views
Homework: $(|H|,|K|)=1 \Leftrightarrow A=(A\cap H)(A\cap K) \forall A\leqslant G$.
Let $G$ is finite group and $H,K$ are normal subgroups of $G$ such that $G=HK$ and $H\cap K=1$ ($1$ is identity element).
Show that $(|H|,|K|)=1$ if and only if for all subgroup $A$ of $G$ we have ...
3
votes
2answers
51 views
What is the order of $(3, 1) + \langle(0, 2)\rangle$ in the quotient group $\mathbb{Z}_4 \times \mathbb{Z}_8 /(0, 2)$?
Let $\langle(0, 2)\rangle$ denote the subgroup generated by $(0, 2)$ in $\mathbb{Z}_4 \times \mathbb{Z}_8$.
How I can find the order of $(3, 1) + \langle(0, 2)\rangle$ in the quotient group ...
3
votes
3answers
148 views
Group theory - subgroups
Let $G=(\mathbb{Z}_n,+)\,$ be a group where $\,n \geq 2$ and $d \neq 0$.
Find all possible values for $d$ and $n$ so that $\{0,d\}$ is a subgroup of $G$.
I know that for subset of $G$ to be a group ...
3
votes
3answers
171 views
group multiplication table
I really looked all over the web and searched for an example I will understand.
I don't understand how to complete a multiplication table! (all examples I found the Identity element was given)
...
2
votes
3answers
59 views
abelian and finite group
G= $Q^+$ (Rational numbers diffrent from zero)
$a*b = ab/2$
I already proved this is a group now I need to prove or disprove that it is abelian and or finite group.
For abelian - from what I ...
5
votes
5answers
155 views
Is $\{-2,2\}$ a group under $a\star b=\max\{a,b\}$?
Lets say $G={-2,2}$ and $a*b=\text{max}\{a,b\}$.
I need to check if this is a group
and if it does than is it abelian or not and finite or not.
Well... first, I'm not sure if this is a group. for ...
-1
votes
3answers
221 views
How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? [duplicate]
How do I find the number of group homomorphisms from the symmetric group $S_3$ to $\mathbb{Z}/6\mathbb{Z}$?
1
vote
1answer
71 views
Central Series - Normal subgroups
$H$, $N$ are subnormal subgroups in the finite group $G$ and $G = H*N$.
Show: $(H*N)^{\infty} = H^{\infty}*N^{\infty}$.
(And $G^{\infty} := \bigcap\limits_{i\geq 0}G^{i}$, and $G^{i+1} = [G, G^{i}]$ ...
0
votes
2answers
66 views
Composite group homomorphism between alternating groups
Let $N$ a non-trivial normal subgroup of $A_n$ and $H = N \cap A_{n-1}$. I would like to show that $A_{n-1} \hookrightarrow A_n \to A_n/N$ is surjective, where $A_n \to A_n/N$ is the canonical ...
