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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Cardinality of stabilizer in $PSL(2,\mathbb{Z}_7)$

Let $v$ be a non-zero vector in $\mathbb{Z}_7^2$ (up to scaling), and let P be the stabilizer of $v$ (up to scaling) in the Projective special linear group $PSL(2,\mathbb{Z}_7)$. Find the Cardinality ...
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2answers
17 views

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$.

Every proper maximal subgroup of a $p$-group $P$ is normal and has index $p$. I tried to search online by I can't get a complete proof. Take $M$ to be maximal and $Z$ to be central subgroup of order ...
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0answers
30 views

Answer TRUE or FALSE to the following statement. [on hold]

Answer TRUE or FALSE to the following statement. The centraliser of $x_{3}x_{2}x_{1}x_{2}x_{1}x_{3}^{-1}$ in $F_{3}$ is $< x_{3}x_{2}x_{1}x_{2}x_{1}x_{3}^{-1}>$. So I was not clear whether this ...
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1answer
26 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
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1answer
46 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
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0answers
17 views

topology notation $G \backslash X$

Suppose there is a group $G$ acting on space $X$. What does the following notation mean? $$G \backslash X$$ Thank you!
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21 views

Finding the kernel of an epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. We define the following epimorphism from $\Lambda$ onto $S_3$: $\theta: \Lambda(a,b) ...
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4 views

Partitioning the set of mappings.

The following is first two steps of an algorithm given from a research paper. I understood the first step. But please explain the second step: what does mean " Rearrange the partition according to ...
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1answer
39 views

Isomorphic or not: two infinite groups [on hold]

The groups $(\mathbb{C}\setminus \left \{ 0 \right \},\cdot )$ and $(\mathbb{R},+)$ are not isomorphic. So I was not clear whether this statement is true or not.
3
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1answer
35 views

$\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$?

What can I use to display the following: $\{(1), (12)(34), (13)(24), (14)(23)\}$ is the only non-cyclic proper subgroup of $A_4$. What I've started to do: list all the elements of $A_4$ and finding ...
3
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2answers
44 views

Prove that if Aut($G$) is the trivial group, then so is $G$? [duplicate]

Let $G$ be a finitely generated group. Show that if Aut($G$) is the trivial group, then so is $G$. I know that if Aut($G$) is the trivial group then $G$ must be abelian but I'm not sure how to ...
0
votes
1answer
25 views

What is an order of an element of a partition"?

I'm reading a paper, in which the set of all 3^3 mappings from {0,1,2} to itself (for instance {001,020,110,121,122}, {002,010,112,011}, {0,1,2}, ...) is partitioned, after which is written two ...
4
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1answer
29 views

Constructing well-defined epimorphism onto $S_3$?

Let $\Lambda$ denote the group with presentation $\langle a,b \mid abab^{-1}a^{-1}b^{-1}\rangle$. Construct an epimorphism from $\Lambda$ onto $S_3$, making sure to check that the function is ...
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0answers
14 views

$U$ subset of $S$, $G$ acts transitively on $S$, show that the subsets $gU$ cover $S$ evenly

I have a finite set $S$ on which a group $G$ acts transitively. Now, I let $U$ be a subset of $S$. I want to show that the subsets $gU$ cover $S$ evenly, meaning every element of $S$ is in the same ...
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2answers
28 views

$f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$.

Let $f(x) \in \mathbb{R}[x]$. Prove that if $z = a + bi$ is a zero of $f(x)$ then $z = a − bi$ is also a zero of $f(x)$. I'm learning about polynomial rings but my book and my instructor never ...
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1answer
12 views

The minimal group with Fitting length three has $p$ section in middle?

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. Assume that for every proper characteristic subgroup $K$ of $G$, Fitting ...
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1answer
52 views

Don't understand the proof of Artin's “Algebra” Ed 1, Prop 5-8.4

I'm reading Artin's Algebra, Edition 1. In Chapter 5 there's proposition (8.4): Let $c_g$ denote conjugation by $g$, the map $c_g(x) = gxg^{-1}$. The map $f: S_3 \rightarrow Aut(S_3)$ from the ...
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1answer
47 views

Can we find a non central element of order 2 in a specific 2-group?

Let $G$ be a non-abelian group of order $2^5$ and center $Z(G)$ is non cyclic. Can we always find an element $x\not\in Z(G)$ of order $2$ if for any pair of elements $a$ and $b$ of $Z(G)$ of order ...
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0answers
31 views

Proof of Cayley's Theorem

This question relates to the link: Cayley's theorem The way I reasoned in showing the map T is a Homomorphism is as follows: Definition: A Homomorphism $\phi: \left ( G,\ast \right ) ...
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1answer
35 views

Can every group be obtained from a choice of Sylow subgroup for every prime divisor?

The question is almost clear from title: If $G$ is a finite group of order $p_1^{n_1}\cdots p_r^{n_r}$ then is it always possible to choose one Sylow subgroup for every prime divisor of $|G|$ ...
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0answers
11 views

Example of lattice of subgroups of quotient group [on hold]

I've studied a theorem that explains what is the lattice of subgroups of a quotient group. The result is the following: Given a group G and a normal subgroup N if we denote by Sub(G) the lattice ...
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1answer
15 views

Is there a direct sum decomposition of the tensor product of two representations of two group elements?

I know that I can decompose $\rho_a(g) \otimes \rho_b(g)$ into $U^\dagger \left[ \rho_c(g) \oplus \rho_d(g) \right] U$. Is there a similar way to decompose $\rho_a(g_1) \otimes \rho_b(g_2)$ into ...
3
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1answer
26 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 ...
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1answer
40 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
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1answer
42 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 ...
4
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1answer
37 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, ...
4
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1answer
60 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 ...
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1answer
50 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
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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
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2answers
45 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 ...
4
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1answer
25 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
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4answers
49 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
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1answer
42 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 ...
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2answers
339 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
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2answers
35 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 ...
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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 ...
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0answers
17 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) ...
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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 ...
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0answers
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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
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3answers
276 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, ...
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2answers
41 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
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2answers
87 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
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1answer
48 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, ...
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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 ...
2
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1answer
33 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
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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, $5$, ...
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2answers
75 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 ...
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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
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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 $[G:K]=1600$. ...