A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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1answer
29 views

If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
4
votes
1answer
91 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
0
votes
1answer
16 views

Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...
5
votes
1answer
54 views

Presentation of a group question

So I know that given a presentation of a group $G$, one can derive from the relations of the group presentation any element in the group $G$ right. However, I do have some confusion. If we take ...
0
votes
1answer
17 views

Properties of generalized characters

I search for generalized characters which are not characters.Also I want to know that why every generalized character is a difference of characters.
-3
votes
1answer
39 views

Generalization of Burnside theorem [on hold]

I probe for understanding the proof of Burnside $ p-q $ theorem: a finite group with a nilpotent subgroup of prime-power index is soluble.
0
votes
1answer
50 views

How many non-isomorphic groups of order 122 are there?

How many non-isomorphic groups of order 122 are there? Let $G$ be a group of order 122.No of Sylow 61 subgroups of order 61=1 and hence it is normal say it is $H$. No. of Sylow 2 subgroups of order ...
2
votes
2answers
61 views

Is group $G$ must abelian, when some condition is given by using exact sequence?

Suppose we are given the following exact sequence of groups where $A$ is an abelian normal subgroup of $G$: $$1 \rightarrow A \rightarrow G \rightarrow Q \rightarrow 1\tag{E}$$ If $G$ is Abelian, ...
0
votes
1answer
35 views

Frobenius Groups [on hold]

Two questions about Frobenius froups: Let $ G $ be a Frobenius group with kernel $ N $ then help me to prove 1- $ |G:N| $ divides $ |N|-1 $, and 2- If $ L\lhd G $then either $ L\leq N $ or $ N\leq ...
3
votes
1answer
58 views

Is there a group homomorphism $f:G\longrightarrow G$ for which $G/\operatorname{Im} f \not\cong\operatorname{Ker} f$? $G$ is finite

Can you find a counterexample to the claim that for all group homomorphisms $f:G \longrightarrow G$, $G/\operatorname{Im} f \cong \operatorname{Ker} f$. Let $G = \mathbb{Z}$; $f(n) = 2n$ is a ...
0
votes
0answers
32 views

Character of transitive finite permutation groups [on hold]

Let $ G $ be a transitive finite permutation group with permutation character $ \pi $ and let $ \chi $ be an irreducible $ \mathbb{C} $-character. I want to know why the degree of $ \chi $ is at least ...
1
vote
4answers
56 views

Prove that $H$ is normal subgroup of $G$

I have a following question. Let $p$ be a prime and let $G$ be a group and $H$ be a subgroup of $G$. $$ G = \left\{ \begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix} : a,b \in \mathbb{Z}_p, a ...
7
votes
4answers
89 views

If $G$ is cyclic then $G/H$ is cyclic?

If $G$ is cyclic, then $G/H$ is cyclic? The proof I got goes like this: $G$ is cyclic, so $G=<g>$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So ...
6
votes
1answer
59 views

Proving that the free group on two generators is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$

I want to show that the free group on two elements $F(\{x,y\})$ is the coproduct $\mathbb{Z}*\mathbb{Z}$ in $\textbf{Grp}$. The idea is to use the universal property of free groups to prove that ...
1
vote
1answer
38 views

If $X \subseteq G$, $\langle X^{G}\rangle$ is normal in $G$

Let $X^{G} = \{gxg^{-1}: g \in G, x \in X\}$, and define $$\langle X^{G}\rangle = \bigcap_{H \in A} H,$$ where $A=\{H \leq G: X^{G} \subseteq H\}$. We wish to show that $\langle X^{G}\rangle$ is ...
11
votes
3answers
726 views

Are there more groups than rings?

It seems pretty clear to me that both of these are at least uncountable (which I think I could prove with some work). It also seems that you should be able to make some diagonal argument about the ...
2
votes
0answers
76 views

$G/Z(G) \simeq S_3 \implies |G'|=3$ [on hold]

Let $G$ be a non-abelian group. Show that if $G/Z(G)$ is isomorphic to the symmetric group $S_3$, then $|G'|=3$. Where $$G' = \langle xyx^{-1}y^{-1} : x,y \in G \rangle$$
0
votes
2answers
30 views

$(h,k).(1,1)=(h\varphi(k)(1),k(1))$ what is $\varphi(k)$?

I am working with: Let $H$ and $K$ be groups and let $\operatorname{Aut}(H)$ be the group of automorphisms of $H$ (under function composition). Suppose also that we are given a homomorphism $\varphi: ...
0
votes
1answer
13 views

If $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ show that $N\lhd G$

Let $G=S_4, N=\{e;(1,2)(3,4);(1,3)(2,4);(1,4)(2,3)\}$ I need to prove that $N\lhd G$ Attempt: $N\lhd G \iff gng^{-1}\in N$ and it is in $N$ for all $g\in G, n\in N$
0
votes
2answers
35 views

If $φ:K → Aut(H)$ is the trivial homomorphism—then the semidirect product is just the ordinary direct product of groups.

I'm working on the question Show that if $φ:K → Aut(H)$ is the trivial homomorphism—i.e., $φ(k) = 1_H$ for every $k ∈ K$— then the semidirect product is just the ordinary direct product of groups. I ...
0
votes
3answers
42 views

Show that $\alpha: G \rightarrow G ,\alpha(g)=g^2$ isomorphism

Let $G$ be a group and $\alpha: G \rightarrow G ,\alpha(g)=g^2$ ,$G$ is abelian group, $|G|=2n+1$ show that $\alpha$ isomorphism Attempt: $\alpha (g_1g_2)=(g_1g_2)(g_1g_2)=(g_1g_1)(g_2g_2)$ ...
2
votes
2answers
39 views

Show that $\psi_a:G\rightarrow G',a\in G,\psi_a(g)=aga^{-1}$ is homomorphism one to one and onto [duplicate]

Let $G$ be a group and $\psi_a:G\rightarrow G',a\in G,\psi_a(g)=aga^{-1}$, I need to show that $\psi_a$ is homomorphism one to one and onto It's not the same question like "Is the conjugation ...
0
votes
1answer
45 views

Irreducible $ \mathbb{C} $-character of a finite groups [on hold]

Let $ \chi $ be an irreducible $ \mathbb{C} $-character of a finite group $ G $ and let $ K $ denote the kernel of the associated representation. If $ \chi $ has degree $ n $, is it true that $ ...
1
vote
0answers
44 views

$ \mathbb{C} $-characters of $ A_5 $

How can we find all of five irreducible $ \mathbb{C} $-characters of $ A_5 $? Precisely how can we construct the character table with the aid of orthogonality relations?
3
votes
1answer
40 views

Isomorphism between two groups of order $p^6$

Let $\mathbb{F}_{p^n}$ denote the finite field of order $p^n$. Let $G$ be the group $$ \begin{Bmatrix} \begin{bmatrix} 1 & a & b\\ 0 & 1 & c\\ 0 & 0 & 1\end{bmatrix}\colon ...
1
vote
1answer
36 views

Facts on $ \mathbb{C} $-characters

My assumption: $ G $ is a finite group & $ \chi $ is a faithful $ \mathbb{C} $-character of $ G $ with degree $ n $ and $ r $ is the number of distinct values assumed by $ \chi $. Now is it true ...
1
vote
1answer
26 views

Homomorphism on U(36)

Question Suppose that $f$ is homomorphism of $U(36)$, $\ker(f) = \{1,13,25\}$, and $f(5) =17$. Determine all the elements that map to 17. What I've tried so far So I've determined that $U(36) = ...
0
votes
1answer
35 views

Is $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{n},n\in \mathbb{N},n>1$ homomorphism?

Let $G$ be abelian group and $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{n},n\in \mathbb{N},n>1$ I need to find if $\varphi$ is homomorphism, and if so to find $ker(\varphi)$ and to tell if ...
5
votes
4answers
97 views

Is $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{-1}$ homomorphism?

Let $G$ be a group and $\varphi:G\rightarrow G $ ,$\varphi(a)=a^{-1}$ I need to find if $\varphi$ is homomorphism, and if so to find $ker(\varphi)$ and to tell if $\varphi$ is one-to-one and\or ...
0
votes
0answers
69 views

Infinite Non Abelian 3- Group [closed]

Does there exist a infinite non Abelian Group whose every non identity element has order 3.
6
votes
3answers
53 views

Semigroup law on points on the curve $f(x) = \frac{1}{x}$

Consider the positive half of the curve $f: \Bbb{R} \to \Bbb{R}, f(x) = \frac{1}{x}$. Let $A = (a,1/a), B = (b, 1/b)$ be any two points on the curve. Draw a line through them Find where this point ...
3
votes
1answer
60 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
1
vote
3answers
51 views

Finite field definition question

My textbook says that $ (\mathbb{Z}_{m},+,*)$ is a field if and only if m is a prime number. However, on Wikipedia it says: "Finite fields only exist when the order (size) is a prime power $p^{k}$ ...
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votes
0answers
26 views

Cover Folner sets by disjoint balls [closed]

Let $G$ be a finitely generated amenable group with a symmetric generating set $S$, and let $r$ be a positive integer. I am looking at the the Cayley graph of $G$ with respect to $S$. Is it possible ...
0
votes
2answers
30 views

Show that $f$ is onto .

Let $f:G\rightarrow G'$ be a non trivial homomorphism, Prove that if $|G'|$ is prime, then $f$ is onto. Attempt: As $f$ is homomorphism, $f(gg')=f(g)f(g')$ for all $g,g'\in G$. If $|G'|$ is prime ...
1
vote
1answer
22 views

Find the Existence and Uniqueness of a Cyclic subgroup

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= ...
2
votes
2answers
57 views

How do i proof that that the map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is homomorpism?

I'm trying to proof that a map $ \varphi: Aut(F_n) \to GL_n(\mathbb{Z})$ is a homomorphism but i can't exactly define which is the function to show that. The map $ \varphi $ is a map with for any ...
0
votes
2answers
31 views

prove that $f$ is one-to-one .

Let $f:G\rightarrow G'$ be a non trivial homomorphism, Prove that if $|G|$ is prime, then $f$ is one-to-one. Attempt: As $f$ is homomorphism, $f(gg')=f(g)f(g')$ for all $g,g'\in G$. If $|G|$ is ...
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votes
1answer
70 views

$ \mathbb{C} $-character table of $ D_{14} $ [on hold]

Is there any reference where I can find the $ \mathbb{C} $-character table of the dihedral group $ D_{14} $?
3
votes
1answer
34 views

simple question on conjugacy classes

if $ \;G = \langle a,b\;|\; a^9 = b^3 = 1, bab^{-1} = a^4\rangle\; $ of order $\;27\;$ Then how would i show that $b$ is conjugate to $ba^3$ I have been fiddling around with this for ages and cannot ...
4
votes
2answers
47 views

Finding the character table of this group

if $ G = <a,b| a^9 = b^3 = 1, bab^{-1} = a^4> $ of order 27 then know the following, that any element can be written as $b^ka^n$ with n $\in [0,8], k\in[0,2]$ and that the 11 conjugacy classes ...
4
votes
1answer
42 views

Orbit Stabilizer problem (I think)

Let $G$ (finite) act transitively on the nonempty set $\Omega$. Show that if $\alpha \neq \beta$ are elements of $\Omega$ then $G_{\alpha}G_{\beta}$ is a proper subset of $G$, where $G_{\alpha}$ and ...
1
vote
0answers
20 views

Isomorphic encryption or homomorphic encryption?

Many encryption functions are said to be homomorphic: http://en.wikipedia.org/wiki/Homomorphic_encryption As encryption functions are invertible, they can be considered one-to-one and onto on ...
1
vote
2answers
51 views

irreducible characters of a group

I am currently attempting a past exam paper and am stuck on the following question for part a) $\mu$ is an irreducible character iff it is equal to the character of an irreducible representation, ...
2
votes
1answer
27 views

Direct decompositions and quotients of abelian groups

Let $G = \langle a \rangle_{27} \oplus \langle b \rangle_{81}$. Find a direct decomposition $G = \langle 10a + 60b \rangle \oplus ?$. Find the elementary divisors of $G/ \langle 3a + 18b \rangle$. ...
1
vote
1answer
70 views

character tables and solubility

I am currently going through a past exam paper for a group theory module and am unable to answer the following section of a question. The copy of my lecture notes doesn't seem to have a section on ...
0
votes
1answer
76 views

Herstein Problem No.7 Page 102

let $G$ be a group of order $30$ .How many non-isomorphic groups of order $30 $ are there? Before doing this I have shown that every Sylow 3 and Sylow 5 subgroup is normal in G and G has a normal ...
0
votes
1answer
26 views

How do I show that the identity element e is contained in the subset H of a group G?

Suppose the set $G$ is an additive group of integers $(G,+)$. For a subset $H$ of the set $G$ to be a subgroup, the subset $H$ must contain the identity element the subset $H$ must be closed under ...
3
votes
4answers
64 views

Topics in Algebra I.N.Herstein Problem 7

Given that if $A$ and $B$ are cyclic of orders m and n and $\gcd(m,n)=1$ then $A\times B$ is cyclic. Using this prove that if $u,v\in \mathbb Z$ then $\exists x$ such that $x\equiv u(\mod m);x\equiv ...
0
votes
2answers
41 views

what does this notation means in group theory?

Quoting from my text, "Observe that if we let $$\mathbb{R}\times \mathbb{R}$$ denote all ordered pairs of real numbers,....." What does the notation "$\times$" means?