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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269 views

What groups can G/Z(G) be?

Let $G$ be a finite group and let $Z(G)$ denote its center. A simple result states that if $G/Z(G)$ a nontrivial cyclic group then $G$ is abelian. Of course if $G$ is abelian then $Z(G)=G$ and ...
1
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1answer
621 views

give the lists of elementary divisors for an abelian group of the specified order and match each list with corresponding list of invariant factors

Give the lists of elementary divisors for an abelian group of the order $270$ and match each list with corresponding list of invariant factors. Why the elementary divisors corresponding to invariant ...
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0answers
43 views

What alternative methods have been proposed for proving the (FS groups) classification theorem?

I am told that John Conway thought that the Classification of Finite Simple Groups Theorem should have been done "inside-out". What sources would be useful in understanding the alternative forms of ...
0
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1answer
178 views

Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$

I am trying to prove the above with the conditions that N is normal to G and $G/N$ is abelian. So then for $Na,Nb\in{G/N}, NaNb=NbNa$. But $NaNb=Nab=Nba$, so then we know $ab=ba$, so G is abelian. ...
1
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1answer
48 views

$[H\vee K:H]\ge [K:H\cap K]$

I'm trying to prove that If $H$ and $K$ are subgroups of a group $G$, then $[H\vee K:H]\ge [K:H\cap K]$ I know that $K\subset HK\subset H\vee K$, this easily implies that $[HK:H]\le [H\vee K:H]$. ...
2
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0answers
53 views

Why does $d_{\alpha}$ divide $\#G$ for $\alpha\in\hat{G}$?

Let $\alpha$ be a unitary irreducible representation of a finite group $G$. Then we have \begin{equation} d_{\alpha}|\#G, \end{equation} where $d_\alpha$ is the degree of the representation and $\#G$ ...
4
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2answers
257 views

understanding $\mathbb{R}$/$\mathbb{Z}$

I am having trouble understanding the factor group, $\mathbb{R}$/$\mathbb{Z}$, or maybe i'm not. Here's what I am thinking. Okay, so i have a group $G=(\mathbb{R},+)$, and I have a subgroup ...
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0answers
47 views

“adding” numbers $\in\mathbb{U\left(2\right)}\times \mathbb{R}^+_0$

Let a unitary number be one that corresponds to a matrix of the form: $$\left( \begin{array}{cc} w+i x & y+i z \\ -e^{i p} (y-i z) & e^{i p} (w-i x) \end{array} \right)$$ This is analogous to ...
3
votes
3answers
100 views

Prove these subgroups are equal

I'm trying to solve this question which I found very complex and difficult due to the amount of details. Let $H, K, N$ be subgroups of a group G such that $H\leq K$, $H\cap N =K\cap N$, and ...
4
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2answers
229 views

Finding the kernel of a homomorphism

I have the groups of nonzero complex numbers and the positive real numbers and the homomorphism $f: \Bbb{C}^{*} \to \Bbb{R}_+$ such that $f(z)= \lvert z \rvert$. I need to find the kernel of f. ...
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1answer
38 views

prove that normality satisfies intermediate subgroup condition

How to prove that normality satisfies intermediate subgroup condition? That is, to prove this: Let $H\leq K\leq G$ be groups s.t. $H \triangleleft G$ ($H$ is normal in $G$), then $H \triangleleft K$. ...
4
votes
1answer
123 views

Reference of a Theorem in Group Theory

A finite group can not be union of conjugates of a proper subgroup. Is this a theorem of "Camille Jordan"? If not, who proved this statement first? Please provide a reference. (I didn't get any ...
4
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2answers
199 views

Order-preserving isomorphism between $\mathbb{R}^n$ and $\mathbb{R}$

Suppose we have a linearly ordered group over $\mathbb Z^n$ where the ordering goes left-to-right, i.e. when deciding if $(x_1,x_2,\dots)<(y_1,y_2,\dots)$ we first check if $x_1< y_1$, if it is ...
3
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2answers
1k views

Determine Whether Two Matrix Groups Are Isomorphic Or Not.

Here is the problem I am having trouble with: Let: $$G = \left\{\begin{bmatrix} a & b\\ 0 & c \end{bmatrix} \in GL(2,R)\right\}$$ $$H = \left\{\begin{bmatrix} a & 0\\ 0 & b ...
3
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4answers
138 views

What math will I need in order to learn Microsoft's UProve?

I'm studying Microsoft's UProve (independent studies at 35 years old) and forget most of the Math I learned in college. I intend to proceed and learn the contents of this chapter of this book but can ...
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2answers
289 views

Showing that $G$ is a group under an alternative operation.

Let $G$ be a group and let $c$ be a fixed elements of $G$. Now, I'm going to define a new operation "*" on $G$ by $a*b=ac^{-1}b$ How do I prove that the set $G$ is a group under *. Thanks for ...
7
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2answers
1k views

When is the group of units in $\mathbb{Z}_n$ cyclic?

Let $U_n$ denote the group of units in $\mathbb{Z}_n$ with multiplication modulo $n$. It is easy to show that this is a group. My question is how to characterize the $n$ for which it is cyclic. Since ...
3
votes
1answer
249 views

Group theory - proof check about index and quotient group

I'm studying Cayley's Theorem on the Humpreys "A Course in Group Theory" and i did not understand a passage in a preposition. (pag 86 Corollary 9.23). It claims: "Let $H \leq G$ with finite index $n$. ...
1
vote
1answer
166 views

If N is a normal subgroup of G,decide whether np(N) | np(G) or np(G/N) | np(G)?

Here np(H) represents the number of sylow-p subgroups of H.Thanks in advance.
3
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2answers
154 views

Number of Abelian Groups of Order 256

I am trying to find the number of abelian groups of order 256. Is the following correct? We may write $256=2^8$ we then know that this may be represented in the form: $C_{n_1}\times.....\times ...
5
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1answer
100 views

Showing that if $N \le G$ is finite minimal normal with every simple homomorphic image abelian, then it is abelian itself

I've been working on the following problem, with no success so far: "Let $N$ be a finite minimal normal subgroup of a group $G$, and suppose $N$ has the property that every simple homomorphic image ...
1
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1answer
51 views

$HK\cap N=H(K\cap N)$

I'm trying to prove If $H$, $K$ and $N$ are subgroups of a group $G$ such that $H\lt N$, then $HK\cap N=H(K\cap N).$ I'm trying sets inclusion to prove it, am I in the right way? I need help. Thanks ...
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2answers
91 views

Conjugation in $S_n$

We have that any element in $S_n$ is generated by the adjacent transpositions $(12),\dots,(n-1,n)$. I am trying to calculate the conjugation of $(ab)\in S_n$ by $\sigma\in S_n$. So if we write sigma ...
4
votes
1answer
188 views

Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
7
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2answers
1k views

If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$

I'm struggling to proof that if $H$ and $K$ are subgroups of a finite index of a group $G$ such that [G:H] and [G:K] are relatively prime, then $G=HK$. I don't know why I can't answer it, because this ...
2
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0answers
187 views

Element of a certain order in multiplicative group of residues

Let's say that $G$ is the multiplicative group of residues mod $p$, where $p$ is prime. I know that the order of an element $g \in G$ is the least $k$ for which $g^k \equiv 1 \mod{p}$. How can we go ...
4
votes
3answers
445 views

finite abelian group satisfying $x^2=e$

I looked but didn't see this question pop up. Not homework as I am graduating on Thursday and took Abstract a year ago. I'm taking the Praxis II and honing my skills. I have good intuition about ...
4
votes
1answer
152 views

Examples of groups with a certain number of Sylow 2-subgroups

Let $G$ be a group of order 50 and $m$ be the number of Sylow 2-subgroups of $G$. What are the possible values of $m$? For each value in your list, give an example of a group $G$ for which $m$ ...
2
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3answers
550 views

If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ .

If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ . With the assumption, I dont know how to start the proof. If there is no non-trivial ...
2
votes
3answers
576 views

All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]

All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $ How could I find every group homomorphism?
3
votes
2answers
1k views

A non-abelian group of order $ 6 $ is isomorphic to $ S_3 $

I know that it is duplicated. But I'm confusing some step of this proof. Please help me. pf) Let $ G $ be a nontrivial group of order $ 6 $. Since $ G $ is non-abelian, no elements in $ G $ have the ...
2
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2answers
221 views

If $G$ is a group, $H,K \leq G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then… [duplicate]

If $G$ is a group, $H$ and $K$ both subgroups of $G$, $K \subseteq H$, $\left[G:H\right]$ and $\left[H:K\right]$ both finite then $\left[G:K\right]=\left[G:H\right]\cdot\left[ H:K \right].$ I am ...
2
votes
4answers
103 views

number of subgroups of order $4$ of $\mathbb Z_4\oplus\mathbb Z_2?$

Without using the property of finite abelian group how to evaluate the number of subgroups of order $4$ of $\mathbb Z_4\oplus\mathbb Z_2?$ Please help ! I can show that $\mathbb Z_4\oplus\mathbb Z_2$ ...
3
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2answers
83 views

Given a topological space $X$, does $H_1(X)=\mathbb{Z}$ imply $\pi(X)=\mathbb{Z}$?

Let $X$ be any topological space with the first homology group $H_1(X)=\mathbb{Z}$ . I claim $\pi_1(X)=\mathbb{Z}$. By Hurewicz, we know that $H_1(X)$ is the abelianization of $\pi_1(X)$. ...
4
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2answers
378 views

Show abelian groups of order 3240?

Show how to get all abelian groups of order $2^3 \cdot 3^4 \cdot 5$. I just started learning this and was wondering how you would do this? Is this correct? $2^3 \cdot 3^4 \cdot 5 = 3240$. Therefore ...
1
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3answers
132 views

If H is a p-group, the order of any H-orbit is a power of p.

This comes from the proof of the third Sylow Theorem in Michael Artin's "Algebra". Let S be the set of Sylow p-groups in a given group G of order $p^em$. Let H be any Sylow group. If we decompose S ...
7
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1answer
84 views

Show $SL(2,\mathbb{Z})$ written as finite product of elements of a particular form

Prove that any element of $SL(2,\mathbb{Z})$ can be represented by a finite product of matrices of the following form. $$\begin{pmatrix}1-ab & a^2\\ -b^2 & 1+ab\end{pmatrix}.$$ We are given ...
0
votes
2answers
844 views

Filling up the Cayley table and finding Self-inverse

The set $G$ is given by $G = \{a, b, c, d, f, g, h, k\}$. $(G, *)$ is a group, with identity $k$, under a certain binary operation $*$. $a * b = c$, $b * a = d$, $f * f = a$, $g * g = b$, $h ...
2
votes
3answers
123 views

Group with an order?

Let $G$ be a group and $g\in G$ with $\operatorname{ord}(g)=40$. Find the order of $a = g^8$, $b= g^5$, and what is $ab$? I know how to solve the proff $\operatorname{ord}(a)=m$, $\operatorname{ord} ...
2
votes
3answers
148 views

Root of multiplicity?

Show if a is a root of multiplicity $n\geq 2\ $, then $f(a) = 0$ and $f'(a)=0.$ I was trying to learn root of multiplicity and saw this question. My TA did not go over it yet but I was wondering how ...
3
votes
1answer
1k views

Conjugacy classes of D2n? [duplicate]

I'm trying to understand this proof and I can follow it up the the second last paragraph where it states what happens if n is odd/even. I don't understand why there is just one conjugacy class when n ...
9
votes
2answers
389 views

Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$

The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup $$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
4
votes
1answer
199 views

Dihedral group as a matrix group

I wish to consider the dihedral group as a matrix group. One way to do that is to consider it as a finite subgroup of $O_2$, a group of orthogonal $2\times 2$ matrices, defined by ...
1
vote
1answer
55 views

How to show that $O(y)\ne 6?$

Let in a group of $G$ order $6$ has an element $x$ of order $3.$ Choose $y\notin\left< x\right>.$ How to show that $O(y)\ne 6?$
2
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1answer
109 views

Properties which are constant on conjugacy classes of a group

Let $\Phi$ be some property which might hold of an element of a group, and say that in every group, $\Phi$ holds for some element $x$ of the group if and only if it holds for all the conjugates ...
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4answers
96 views

In a finite cyclic group there is only one subgroup of order $k$.

If $G$ is a cyclic group of order $n$ and $k|n$, then $G$ has a subgroup of order $k$, it's easy to prove, but what I'm having troubles to show is there is only one subgroup of order $k$. I need help ...
1
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2answers
121 views

If $G$ is an abelian group of order $p^n$?

If $G$ is an abelian group of order $p^n$, $p$ a prime and $n_1\geq n_2\geq \cdots \geq n_k > 0$, are the invariants of $G$, show that the maximal order of any element in $G$ is $p^{n_1}$. In my ...
0
votes
1answer
55 views

Need to show that order of orbits under group action is non-trivial and intersection of two p-groups is a proper subgroup

I'm working my way through the second and third sylow theorems in my book. Here's the relevant bit: We have a group $G$ of order $p^\alpha m$ where $p$ does not divide $m$. We have that $Q$ is a ...
3
votes
2answers
60 views

Subgroups that can be proven?

How do you show all subgroups of $A_5$ have order less or equal than 12? I know you can use this lemma: If G is a finite group, and H does not equal G is a subgroup of G such that oG $/|/$ i(H)! ...
0
votes
1answer
78 views

A question on Cosets [duplicate]

Let $G$ be a group and $H$ , $K$ be subgroups of $G$ such that $[G:H]$ and $[G:K]$ are finite. Then is it true that $[G:H∩K]$ is also finite ?