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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3
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1answer
30 views

Show the intersection of six subgroups of order $24$ is normal in $G$? [duplicate]

Let $G$ be a group with exactly six subgroups of order $24$. Show that the intersection of these six subgroups is normal in $G$. My thought is that if we can show these six subgroups are normal ...
1
vote
1answer
44 views

How many distinct patterns exist for a 5x5 grid by filling 3 colors?

Using 3 colors to fill in a $5\times5$ grid (you don't have to use all colors), then how many distinct patterns exist? The "distinct" means we have to consider the symmetry. Any effective approach is ...
2
votes
1answer
49 views

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$? [duplicate]

Show that if a group $G$ has odd order, any subgroup $H$ of index 3 in $G$ is normal in $G$. I think this is equivalent to the following: Let $H$ and $K$ be subgroups of a group $G$, with $K ...
5
votes
1answer
43 views

Group with exactly six Sylow 5-subgroups?

Give an example of a group with exactly six Sylow 5-subgroups. I think $A_5$ works because it has 6 subgroups of order 5: $\langle(12345)\rangle,\langle(12354)\rangle, \langle(12435)\rangle, ...
5
votes
1answer
67 views

Example of a non-abelian quotient of a non-abelian finite group?

Give an example of a non-abelian quotient of a non-abelian finite group. This should be fairly simple but I am drawing a blank. I can think of plenty of non-abelian, finite groups but no ...
1
vote
1answer
58 views

What is the next term in the jumping champions?

I am interested in the jumping champions of the number of groups of order $n$, where $n$ is cubefree. After calling LoadPackage("cubefree"); GAP gives the ...
1
vote
1answer
43 views

For which $n$ in $\mathbb Z/n\mathbb Z$, $\bar{2}$ has a multiplicative inverse.

I am looking for which $n$ in $\mathbb Z/n\mathbb Z$, $\bar{2}$ has a multiplicative inverse. Attempt: I know that I need a $\bar{k}$ such that $\bar{k}$$\bar{2}$ $= \bar{1}$. I believe that the ...
7
votes
2answers
50 views

With the exception of $\mathbb{Z}$, every infinite abelian group contains a subgroup isomorphic to $\mathbb{Z}^2$?

With the exception of $\mathbb{Z}$, every infinite abelian group contains a subgroup isomorphic to $\mathbb{Z}^2$. Is this statement true? I don't have much experience working with non-finite ...
5
votes
1answer
36 views

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$?

The centralizer of $x_3x_2x_1x_2x_1x_3^{-1}$ in $F_3$ is $\langle x_3x_2x_1x_2x_1x_3^{-1}\rangle$. The centralizer of an element in $F_3$ is the set of elements of $F_3$ that commute with that ...
5
votes
4answers
51 views

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$?

Let $H$ and $K$ be subgroups of $G$ such that $H \cap K = \{e\}$. Then $H \cup K$ is a subgroup of $G$. I know that $H \cup K$ is a subgroup of $G$ if and only if $H \subseteq K$ or $K ...
0
votes
1answer
43 views

Can a triangle free graph represent a Group?

Some facts are- Group can be represented by a graph. Quasi Group can be represented by Latin Square matrix, thus by a Latin Square graph. Group Isomorphism $\leq_p$ Graph Isomorphism. Under this ...
1
vote
2answers
342 views

Does there exist a p-group of order 99?

Does there exist a p-group of order $99$? We can first observe that $99=3^2 \times 11$. I then believe we need to apply Sylow's theorem but I am not sure how, exactly. How can I prove existence ...
3
votes
2answers
36 views

Order of $G$ is 35, are there elements of order 5 and 7 in G?

I have a group of order 35 and I want to know if it contains elements of order 7 and 5. I know that it does and there is a proof that is much longer, but I wanted to know if the following worked to ...
0
votes
1answer
22 views

Group action on coset space is continuous

I found this exercise in various places, but I could not find the answer anywhere. As I am quite new to topology, I would appreciate any help. Let $G$ a topological group and $H$ a subgroup. Let the ...
2
votes
0answers
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
votes
0answers
38 views

unfaithful group action [on hold]

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 ...
-1
votes
1answer
42 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
278 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 [on hold]

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
55 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, ...
2
votes
0answers
26 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
16 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
56 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
78 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
43 views

Finding subgroup [on hold]

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
vote
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
votes
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
28 views

Cardinality of Infinite Symmetric group [on hold]

How to show $|\mathrm{Sym}(\Gamma )|=2^{|\Gamma |}$ if $|\Gamma |$ is infinite?
2
votes
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
15 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
vote
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. [on hold]

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
votes
0answers
31 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
58 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 ...
1
vote
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
37 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
votes
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
49 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
64 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
130 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
80 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 ...