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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Why is $U(10)\not\approx U(12)$?

I'm trying to understand this example on isomorphism but failing to do so. I know that if two groups have a differing number of elements of each order they are not isomorphic. I assume this is what ...
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1answer
77 views

$GL_n(\mathcal{F})$ is a finite group if and only if $\mathcal{F}$ is finite

Show that $GL_n(\mathcal{F})$ is a finite group if and only if $\mathcal{F}$ has a finite number of elements. These are my thoughts. The order of the group $GL_n(\mathcal{F})$ is ...
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1answer
62 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
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0answers
38 views

Nontrivial homomorphisms from G to T

Let $G$ be a compact metric abelian group. $T$ be the circle group. Let $\mathcal{A}$ be the set of all finite linear combinations of continuous homomorphisms from $G \to T$. I want to show ...
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3answers
55 views

Explain why D3 cannot be a subgroup of D8

To be a subgroup, a subset of a group must satisfy the group axiom but in this case, I do not see how the group axiom plays a part. Could someone explain to me why the above question is true?
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1answer
23 views

What is the order of a rotation group

I was recapping some question on groups for my upcoming exams and chanced upon a question asking for the order of a rotation group $$C_{8}$$. Is the order of a rotation group 2n or $$\frac{2 ...
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2answers
56 views

Quotients of Solvable Groups are Solvable

I just proved that subgroups of solvable groups are solvable. So given that $G$ is solvable there is $1=G_0 \unrhd G_1 \unrhd \cdots \unrhd G_s=G$ where $G_{i+1}/G_i$ is abelian and for $N$ a normal ...
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37 views

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. [duplicate]

Let $G$ be a group containing exactly $2n$ elements, $n\ge1$ integer. Prove that, $\exists$ $x\neq e$ such that $x^2=e$ where $e$ represents the identity of $G$.
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37 views

Groups and morphisms making a lattice structure

(1) Let X and Y be groups. Write mono:X->Y and epi:X->Y (resp.) to denote the existence of a monomorphism or epimorphism from X to Y (resp). mono:X->Y would mean that there is a subgroup of Y ...
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1answer
35 views

What does it mean to find elements in $S_9$ that are “not cycles”?

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!
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2answers
41 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
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1answer
38 views

Intuition on Hall subgroups and solvability

I see there are many questions on Hall subgroups, but I can't find one that answers my question. Let $G$ be a finite group. A subgroup $H<G$ is a Hall $\pi$-subgroup if its order is the product of ...
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1answer
41 views

group theory - subgroup question (closure)

Q) Define the relation ρ on R by the rule: ∀x, y ∈ R, x ρ y if and only if ∃n ∈ Z such that y = x + nπ. let G denote the set of all equivalence classes under ρ. Let + : G × G → G be defined as [x] + ...
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3answers
47 views

How to determine non trivial homomorphisms [closed]

I am trying to understand and it doesn't make any sense to me: How can I determine if there are any non trivial homomorphisms between groups or rings? How do I find them? and once I found them, how ...
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1answer
37 views

Prove that every element $S \in SO(n)$ is a product of even numbers of reflections

Prove that every element $S \in SO(n)$ is a product of even numbers of reflection. I proved that for SO(2),SO(3) and 2 reflections. How is it with SO(n)?
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44 views

Schur multiplier of “large” groups.

Let $G$ be a finite group and let $M(G)=H^2(G,\mathbb{C}^*)$ be its Schur multiplier. For "small" groups I can compute the Schur multiplier by hand in terms of corresponding roots of unity. However, ...
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0answers
47 views

What is the diameter of the Cayley graph of $S_n$ w.r.t the generator $\{(12),(23),…,(n−1n)\}$?

Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of ...
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1answer
36 views

Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1 [duplicate]

$(G,*)$ is an abelian group and $a,b$ are elements of G. Let $n=ord(a)$ and $m=ord(b)$ in $G$. Show that the order of $a*b$ is equal to $nm$ if $gcd(n,m)=1$. I have already proved that $ord(a*b)|nm$. ...
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32 views

Example of a metric group over $\mathbb{R}_0^+$

Do you know an example of a function $d:\mathbb{R}_0^+\times\mathbb{R}_0^+\to \mathbb{R}_0^+$ for which the following properties hold? Or can you prove this does not exist? There exists an $e\in ...
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0answers
64 views

Fundamental Group of a Surface [closed]

I came across the term "fundamental group of a surface" while reading a paper, and I'm not sure what it it all about. As well, what is understood by the generators of the fundamental group of a ...
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2answers
19 views

Construct non trivial group endomorphism (rigid motion group)

My question, in it's general formulation, is : is there a way to construct non trivial group endomorphism other than conjugation ? Now for my specific needs, I wont to find some endomorphism other ...
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1answer
56 views

What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
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1answer
19 views

Finding the pattern in number of (next and up) nearest neighbours of $N$ points on a circle

The question is rather vague (I couldn't phrase it more clearly though) so let me explain. Say you have a circle, and you put four equidistant points on it. You then have two types of neighbours: ...
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Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is…

Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n\mathbb Z$. Prove that either every element of $H$ is even or exactly half of its elements are even.
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36 views

Number of complements

If $G$ has a normal Hall subgroup $U$ then $U$ has a complement $V$ in $G$ and all of these complements are conjugate. Can we say something about the number of complements? Or in other words: How ...
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1answer
62 views

How does one define a group with commutative diagrams?

I am currently working through McLarty's book on Elementary Categories, Elementary Toposes. In Chapter 3, he considers a group as an object in a category with a unit map, a multiplication and an ...
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31 views

On Thompson conjecture

Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Consider $N(G)=N(A_n)$ and $n/2<p,q<n$, where $p,q$ are prime number. Moreover, assume that Sylow $p$-subgroups and Sylow ...
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37 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
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27 views

What is subgroup lattice of GL$(n,\mathbb F_q)$?

I am trying search for subgroup lattice diagrame for the general linear group GL$_n(\mathbb F_q)$ but could not find any thing in the net. Can some one help me by providng some link on it ? thank ...
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29 views

What are the generators of the coset of SU(2)/U(1)?

I need to understand the coset, $SU(2)/U(1)$ for the fundamental representation. How would I go about doing so? From my understanding of coset it means that any transformation of $SU(2)$ mod a $U(1) ...
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1answer
40 views

Algebraic Peter-Weyl theorem in the case of $G=SL_2$.

The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible ...
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1answer
51 views

Subgroup of symmetric group Sn

Suppose $G$ is a transitive subgroup of $S_n$ such that it there exist $\sigma, \tau \in G$ such that $\sigma$ is an $n-1$-cycle and $\tau$ is a transposition. Prove that $G = S_n$ I just don't ...
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49 views

On conjugacy class size of an element and its order.

Let $G$ be a finite group and $x\in G$. Also we denote the conjugacy class of $x$ in $G$ by $x^G$. I want to know if there is any relation between $|x|$ and $|x^G|$? Suggestions would be appreciated.
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14 views

Conjugate closure and factor group

Let $N\unlhd K$ be a normal subgroup of a given group $K$ and let $$q:K\to K/N$$ be the natural quotient map. Let $A\subseteq K$ be a subset of $K$ and let the conjugate closure of $A$ in $K$ be ...
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1answer
55 views

How to compute the automorphism group of split metacyclic groups?

I am trying to calculate the automorphism group of an affine subgroup $$G=\mathbb{Z}_p\rtimes\mathbb{Z}_{k}\leq\text{AGL}(1,p).$$ One might guess $\text{Aut}(G)=\text{AGL}(1,p)$. And this matches ...
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1answer
25 views

action of a subgroup on a metric space

If $G$ acts properly and cocompactly by isometries on the metric space $X$ and if $H$ is a subgroup of $G$. Does $H$ act properly and cocompactly by isometries on a subspace of $X$?
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1answer
50 views

Prove the group is a direct product [closed]

Let $G$ be an abelian group of finite order $n = mk$ with gcd$(m,k) = 1$. For $r=m,k$, let $G(r) = \{g \in G: g^r = 1 \}$ . Prove that $G = G(m) \times G(k)$.
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1answer
23 views

Finding cosets of a quotient group: List the cosets of $HN/N$

In the group $\Bbb Z_{24}$, let $H=\langle 4\rangle $ and $N=\langle 6\rangle $ 1) List the elements of $HN$. I found $HN=\{0,2,4,\cdots,22\}=\langle 2\rangle$ 2) List the elements of $H\cap N$. I ...
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25 views

How many symmetries does the Cauchy-Schwarz inequality have?

The symmetries of the Cauchy-Schwarz inequality $(x_1^2 + \cdots + x_n^2)(y_1^2 + \cdots + y_n^2) \ge (x_1y_1 + \cdots + x_ny_n)$, as a subgroup of the symmetric group on the $2n$ letters $x_1,\ldots, ...
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3answers
56 views

Is $H\cup K$ a group?

If $H$ and $K$ are subgroups of $G$ is $H\cup K$ also a subgroup of $G$? We have identity for sure(since it is in $H$ or $K$), associativity is absorbed. Thus we only need to see if inverses and ...
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What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
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1answer
62 views

Showing that the product $x*y := \frac{x+y}{xy+1}$ is a group operation on $(-1, 1)$ [duplicate]

I need to show that the following is an abelian group: $$x*y = \frac{x+y}{xy+1}$$ on the set $\{x \in \Bbb R \,|\, -1 < x < 1\}$. I have been working on this problem, trying to show ...
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118 views

Difference between centralizer and center groups?

This is probably stupid question, but I can't see the difference between the two subgroups: $$C_G(A)=\{g\in G| gag^{-1}=a,\forall a\in A\}$$ $$Z(G)=\{g\in G| ga=ag,\forall a\in G\}$$ Is the ...
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0answers
50 views

Short exact sequence of groups

Let $1\to N\xrightarrow{\text{i}} G\to H\to 1$ be a short exact sequence of groups. Let $f:G\to N$ be a group homomorphism such that $fi: N\to N$ is an isomorphism. What are some "good" things we can ...
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0answers
22 views

Graph Relatives for Tessellation of the Hyperbolic Plane

I'm trying to get into the theory about the Moduar group. Among the "Paracompact hyperbolic uniform tilings in [∞,3] family" in the section "Tessellation of the hyperbolic plane" I found the Order-3 ...
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1answer
17 views

Extra-Special $p$ group and complement

Let $G$ be an extra special $p$ group of order $p^{2n+1}$, $n\geq 2$. Does $[G,G]$ necessarily have a complement in $G$? I dont think so, but I am not sure. Sorry this should be a very silly ...
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1answer
51 views

Every group of order $35$ is cyclic? [duplicate]

Prove that every group of order $35$ is cyclic. Now, the subgroups of this are ones whose orders divide the order of this group(by lagrange), these are of prime orders $7$ and $5$. and I guess ...
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36 views

Number of groups of a certain order given: 1) finitely generated abelian, 2) subgroups, 3) not necessarily finitely generated

1) How many finite abelian groups are there of order $1000$? Well via the fundemental theorem of finitely generated abelian groups, we look at the factorisation for $1000$. $1000=2^3*5^3$ and there ...
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4answers
88 views

How do I calculate $2^{47} \pmod{\! 65}$?

I'm trying to calculate $2^{47}\pmod{\! 65}$, but I don't know how... I know that: $65=5\cdot 13$ and that: $2^{47}\equiv 3 \pmod{\! 5}$ and $2^{47}\equiv 7\pmod{\! 13}$... (I used Euler) But ...
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0answers
58 views

The Group of rotations of a cube in space.

Am currently working on the Group of rotations of a cube in space. i have identified the 24 rotational symmetries of a cube that forms a group. And i am kind off trying to show that these symmetries ...