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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20 views

Day's fixed point theorem

Day's fixed point theorem (Theorem 1.3.1; Lecture on amenability; Volker Runde) Let $G$ be a locally compact group. The following are equivalent: $G$ is amenable. If $G$ acts (from left side) ...
-5
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0answers
38 views

unfaithful group action [closed]

Let the group $G=GL\left ( n,\mathbb{F} \right )$ and $\Omega$ be the set of all 1-dimensional subspace of $\mathbb{F}^{n}$ Let $\left \langle V \right \rangle \in \Omega$. Define $\left( \langle ...
-2
votes
1answer
46 views

Is $G/N$ isomorphic to $\mathbb R ?$

$$G=\left\{\begin{bmatrix} a & b \\ 0 & \ \ \ \ a^{-1}\end{bmatrix} : a,b \in \mathbb R;a>0 \right\}$$ $$ \hspace{-1.1in}N=\left\{\begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} :b\in ...
4
votes
0answers
22 views

Determine the center of the group $GL_n(\mathbb{R})$? [duplicate]

Determine the center of the group $GL_n(\mathbb{R})$. The center of a group $G$ is the set of elements that commute with every element of $G$. I think the answer is $Z(GL_n(\mathbb{R}))=\{\lambda ...
6
votes
3answers
284 views

Prove that any two cyclic groups of the same order are isomorphic?

Prove that any two cyclic groups of the same order are isomorphic. Let the groups be $G,H$ with order $k$. Let $G=<a>$ and $H=<b>$. Thus we have $|a|=|b|=k$ and by definition, ...
-3
votes
2answers
43 views

Automorphism of a group is a group action [closed]

Let G be a group and let $\Omega$ be a set. Then, the $Aut\left ( G \right )$ acts on $\Omega=G$ How can I show that this is true? Thank in advance.
4
votes
2answers
93 views

What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such ...
5
votes
1answer
58 views

Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
3
votes
0answers
34 views
+50

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
1
vote
1answer
17 views

Why doesn't the coset (1,4,2,3)K belong to the Quotient group

I had been given the following question and answer ( in the image) However i do not understand, why for example: (1,4,2,3)K does not belong the the quotient group? Is there any faster way of ...
4
votes
2answers
60 views

What's the order of a semigroup?

For a group the order, of an element is the smallest positive integer m such that a^m = e. But what's the order of an element of a semigroup? Or there isn't anything like that?
3
votes
1answer
40 views

Show that any group of order 20 is not simple?

Show that any group of order $20$ is not simple. Denote the group $G$. It seems intuitive to state first that $20=2^2 \times 5$. Sylow's theorem then states that since a prime number, $5$, ...
1
vote
2answers
80 views

Is there some geometric intuition for the quotient $G/Z(G)$, where $G=GL_n(\mathbb{R})$?

Let $G=GL_n(\mathbb{R})$ be the $n$th general linear group. Its center $Z(G)$ is given by all scalar matrices $aI$ with nonzero determinant. How can I get an intuitive picture of $G/Z(G)$? I know that ...
-1
votes
1answer
44 views

Finding subgroup [closed]

Find a subgroup $\mathrm{H}$ in group $\mathbb{Z^2}$ so that $\mathbb{Z^2}/\mathrm{H} \simeq \mathbb{Z_6} \times \mathbb{Z_{10}} \times \mathbb{Z_{15}}$.
1
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1answer
23 views

Formal product of cycles in a permutation group

In Dixon's book Permutation Groups, there is a sentence saying that in a symmetric group $Sym(\Omega)$, "the second common way to specify a permutation is to write $x$ as a product of disjoint ...
0
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1answer
17 views

Intersection of two subgroups with given information

suppose we know that $G$ is a finite group with order $43200$ and suppose that $H$ is a subgroup of $G$ with order $80$. Furthermore, assume that $K$ is also a subgroup of $G$ such that $[G:K]=1600$. ...
0
votes
1answer
31 views

Cardinality of Infinite Symmetric group [closed]

How to show $|\mathrm{Sym}(\Gamma )|=2^{|\Gamma |}$ if $|\Gamma |$ is infinite?
2
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0answers
17 views

Groups acting on Riemann Surfaces and Automorphic Function

Consider the following paragraph from a book of Magnus on Combinatorial Group Theory. ... the simple group $G_{168}$ of order $168$ acts on a genus $3$ surface, is important for the theory of ...
0
votes
1answer
24 views

showing a set is a subgroup of a normaliser

Let $H$ be a subgroup of a group $G$ and defined $N_{G}\left ( H \right )=\left \{ g \in G \mid g^{-1}Hg=H \space\ \right \}$ Show that $H$ is a normal subgroup of $N_{G}\left ( H \right ).$ The ...
2
votes
2answers
16 views

Show normal subgroup $N$ is a subset of $\ker\phi$ if $\psi$ is well-defined

Let $\phi:G\rightarrow K$ be a group homomorphism. Suppose $N$ is a normal subgroup of $G$. Show that if $\psi:G/N\rightarrow K$ defined by $\psi(gN)=\phi(g)$ for all $g\in G$ is well-defined, then ...
0
votes
1answer
6 views

Specific condition for a map to be isomorphism

Let $G=\left ( \mathbb{R} \space\ \text{where} 0 \notin \mathbb{R},\cdot \right )$ and let r be a positive integer. Define $\phi:G\rightarrow G$ $x \mapsto x^{r}$ Show that $\phi$ is an ...
0
votes
0answers
17 views

$\Phi : \mathbb{Z}/mn\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}$ is an Injective application

Let m and n be distinct prime numbers greater that 1. I am trying to show that the following application in an isomorphism: $\Phi : \mathbb{Z}/mn\mathbb{Z} \rightarrow \mathbb{Z}/m\mathbb{Z} \times ...
1
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2answers
46 views

$\bigcup_{n=1}^{\infty}G_n $ is a multiplicative subgroup of the group $\mathbb C\setminus\{0\}$

If $G_n$ is the set of all $n$-th roots of unity for $n\in \mathbb{N}$ then show that $\bigcup_{n=1}^{\infty}G_n $ is a multiplicative subgroup of the group $\mathbb{C} \setminus \{0\}$, the order ...
-2
votes
1answer
12 views

If $\psi$ and $\phi$ are homomorphisms show that $\Theta$ is a homomorphism. [closed]

If $\phi: G \rightarrow K$ such that $g \mapsto \phi(g)$ is a homomorphism. If $\psi: G \rightarrow H$ such that $g \mapsto \psi(g)$ is a homomorphism. Show that $\Theta : G \rightarrow K \times H$ ...
0
votes
1answer
15 views

Order of an element in an external direct product

Consider $\mathbb{Z}_{4}\times \mathbb{Z}_{4}=\left \{ 0,1,2,3 \right \}\times \left \{ 0,1,2,3 \right \}$ The element $\left ( 2,0 \right )$ is of order 2 but I cannot figure out why. $2=LCM\left ...
0
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0answers
32 views

Center of $G_1$ x $G_2$ is $\mathbb{Z_{1}}$ x $\mathbb{Z_{2}}$

If $G_1$ and $G_2$ are groups and $\mathbb{Z_{i}}$ is in the center of $G_i$, is there a particular reason that the center of the product $G_1$ x $G_2$ is $\mathbb{Z_{1}}$ x $\mathbb{Z_{2}}$. I'm ...
0
votes
1answer
59 views

Proving Lagrange's theorem with homomorphisms

Let f:G-->H be a homomorphism, where G is a finite group with identity e1 and H is a finite group with identity e2. Prove that the order of f(g) is a divisor of g for all g in G. So I know that ...
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0answers
12 views

$x$ in group G with order $r$, $y$ in group $G'$ with order $s$ what is the order of $(x,y)$ in $G$ x $G'$

I have an element $x$ of order $r$ in a group $G$ and an element $y$ in group $G'$ of order $s$. Is the order of $(x,y)$ in the product group $G$ x $G'$ $lcm(r,s)$? Thoughts: I think that this is ...
5
votes
1answer
38 views

Given $P,Q$ with prime order, prove $P \cap Q$ is trivial group?

Suppose $P,Q \leq G$ both have prime order, with $P \neq Q$. Prove that $P \cap Q$ is the trivial group. I think Sylow's theorem applies here but I feel like there is not enough information to ...
0
votes
1answer
23 views

An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
7
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1answer
29 views

Possibilities for a group $G$ that acts faithfully on a set of objects with two orbits?

A group $G$ acts faithfully on a set $X$ of 5 objects. The action has two orbits: one of size 2, and one of size 3. What are the possibilities for the group $G$? I think I should apply the ...
1
vote
0answers
54 views

Upto which prime is the calculation of $gnu(2^4\cdot3^4\cdot…\cdot p^4)$ feasible?

The determination of $gnu(n)$, the number of groups of order $n$ upto isomorphism, is very hard in general. But if no large powers are involved, it should be possible for relatively large numbers. ...
8
votes
1answer
68 views

Automorphisms of infinite abelian groups

It is well-known that the map $Aut$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $Aut(S_n) \cong S_n$, $Aut(D_4) \cong D_4$ and if $G$ is a finite non-abelian simple ...
4
votes
3answers
131 views

If $G$ is a finite group and $H \subset G$ is closed, must $H$ be a subgroup?

Supposed theorem (from online notes): If $(G,*)$ is a finite group and $H\subset G$, $H$ is non-empty and $H$ is closed under $*$ then $(H,*)$ is a group. The proof given is a real mess, but, after ...
0
votes
3answers
81 views

What is the one-one and onto mapping?

I'm reading the following paper, and here I can't understand the first step of the algorithm. Please explain what is the one-one and onto mapping? And how he gets these tables ? Commutative ...
4
votes
2answers
119 views

Application of first isomorphism theorem

Let $G$ and $K$ be groups and let $G\times K$ be the direct product of these two groups. Find a normal subgroup $N$ such that $(G\times K)/N\cong G.$ I think I need to use the first isomorphisms ...
1
vote
1answer
22 views

Show that $N \lhd G \times H \not \Rightarrow N = N_1 \times N_2$ with $N_1 \lhd G$ and $N_2 \lhd H$

I'm trying to prove the following assertion: Show that $N \lhd G\times H \not \Rightarrow N = N_1\times N_2$ with $N_1 \lhd G$ and $N_2 \lhd H$ What I tried to do is find $(n_1;n_2) \in N$ such that ...
0
votes
1answer
39 views

finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6

If I have a finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6, is there anything special about G that we can infer? Would the order of $G$ be 30?
5
votes
1answer
38 views

A problem in group theory_dsom [closed]

Let $H$ be a group of integers modp, under addition, where $p$ is a prime number. Suppose that $n$ is an integer satisfying $1 \leq n \leq p$, and let G be the group $ H \times H \times \cdots \times ...
0
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0answers
24 views

Show that There exists a canonical injective homomorphism between $G$ and $G\times H$

Let $(G; *)$ and $(H; \cdot)$ be two groups. The product $G\times H$ is defined by: $G\times H:= \{(g;h)\mid \forall g \in G\text{ and }\forall h \in H\}$ Show that that there exist a canonical ...
0
votes
1answer
25 views

Order in quotient group $G/H$ is not the same in $G$?

$H$ is a normal subgroup of $G$ and $p$ is prime, then $$ord_{G/H}(gH) = p \Rightarrow \exists m \in \mathbb{N}\backslash\{0\}: ord_G(g) = mp$$ Can someone explain why $ord_G(g)$ isn't just $p$?
0
votes
0answers
16 views

Hall subgroup containing all normal $\pi$-subgroups

Let $G$ be a finite group. If $H\leq G$ is a Hall $\pi$-subgroup, show that $H$ contains every normal $\pi$-subgroup of $G$. This is question is proved in some notes of mine. It starts of by letting ...
4
votes
1answer
60 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
4
votes
3answers
56 views

How can I show that $G$ is non abelian of order 20?

Problem says: Let $G=\langle x,y|x^4=y^5=1,x^{-1}yx=y^2\rangle$. Show that $G$ is nonabelian group of order 20. To show it, I tried to turn $x^ny^m$ into $y^kx^l$ for some $k,l$. Since I have ...
1
vote
1answer
33 views

If G contains a normal subgroup $N \cong \mathbb{Z}_2$ and $G/N \cong \mathbb{Z}$, then $G\cong \mathbb{Z}\times \mathbb{Z_2}$.

If G contains a normal subgroup $N \cong \mathbb{Z}_2$ and $G/N \cong \mathbb{Z}$, then $G\cong \mathbb{Z}\times \mathbb{Z_2}$. I'm trying to create an isomorphism $\phi : G \rightarrow ...
0
votes
0answers
21 views

Motivation for proving any nontrivial normal subgroup of $A_5$ has a 3-cycle?

This question is taken from M.A. Armstrong's book Groups & Symmetry. Question 15.12. This aim of this problem is to introduce the concept of a simple group. It asks us to first work out the ...
0
votes
1answer
27 views

Let $H,K$ be subgroups of $G$. Show: $H \trianglelefteq K \Rightarrow K\subset N_G(H)$.

Let $H,K\leq G$. Show that: $$H\trianglelefteq K\Rightarrow K\subset N_G(H).$$ How can I show that a subgroup is normal of a subgroup? For the last part, I've found a proof.
0
votes
2answers
28 views

Order of Group of 2*2 matrix [duplicate]

Let G be the group of 2*2 matrices [ a b ; c d] where a,b,c,d are integers modulo p, p is prime number, such that ad-bc≠0. G forms group under relative to matrix multiplication. What is o(G)? Let H ...
0
votes
0answers
33 views

Is the following function a homomorphism?

Is the following function a group homomorphism? $f:G\to G'$, where $G=(\Bbb{R},*)$ and $G'=(\Bbb{R}^+,o)$, and $f(x)=e^x$.
3
votes
1answer
166 views

What exactly is a p'-group?

In the context of finite group theory I understand that $p$-group is a group whose order is a power of $p$ ($p$ a prime number) but I am unclear on the exact definition of a $p'$-group. This ...