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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591 views

Show two groups are isomorphic

I need to show two groups are isomorphic. Since I know they are of the same order, would finding an element that generates the other elements, in both groups, suffice to show that they are isomorphic? ...
23
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1answer
561 views

“Efficient version” of Cayley's Theorem in Group Theory

I'm considering finite groups only. Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$. I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
2
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1answer
154 views

Subgroup and index in $\mathbb{Z}^2$

Which of the following are finite index subgroups in $\mathbb{Z}^2$? What is the index?? $H=\{(x,y) \in \mathbb{Z}^2 \text{ s.t. } x+y=1\}$ $H=\{(x,y) \in \mathbb{Z}^2 \text{ s.t. } x+y=0\}$ ...
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4answers
80 views

Question about homomorphism of cyclic group

if $\varphi: G\to H$ is homomorphism. How do I prove that if $a\in G$ have finite order so $\varphi(a)$ had finite order to, and that:$$ord(\varphi(a))\mid ord(a)$$ Thank you!
4
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1answer
85 views

If an $H\le G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of at least dimension $d$.

Let $H$ be a subgroup of a group $G$, and let $\rho :H\to GL(V)$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is at least ...
2
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1answer
236 views

Irreducible representations and invariant subspace

We are given a representation of $S_3$ on vector space V. $x$ and $y$ are usual generators of $S_3$. If a is a non-zero vector in $V$ such that $b=a+xa+x^2a$ and $c=a+ya$. Show that $V$ contains a ...
2
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1answer
109 views

With the branching rules of subalgebra, how can I write down explicit matrix elements for a representation?

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
2
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2answers
170 views

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$

How Many Cyclic Subgroups of order $10$ are there in $\mathbb{Z}_{100}\oplus\mathbb{Z}_{25}$? I have calculated that there are $24$ elements of order $10$ I know that in a cyclic subgroup of order ...
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1answer
39 views

Can someone check if I have found all the cyclic subgroups of $D_3$

I have been asked to find all the cyclic subgroups of $D_3$ and therefore determine if $D_3$ has a non-cyclic subgroup. So I have set $D_3$=$G$ and therefore: $G=${$e,a,a^2,b,ab,a^2b$} The cyclic ...
0
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1answer
86 views

π-separable group and subnormal series

I want to show that if $G$ has a subnormal series, then $G$ is $\pi $-separable group. It is enough to show that ${N_i} \triangleleft G$ for every ${N_i} \triangleleft {N_{i + 1}}$ in series. But I ...
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1answer
80 views

Example of $A \le G$ solvable, $B \lhd G$ solvable, but $AB$ is not solvable

We'll denote $A \lhd G$ for $A$ a normal subgroup of $G$; and $A \leq G$ to mean $A$ is a subgroup of $G$. I don't know if it's a tricky question. But it seems strange to me. The question asks for an ...
2
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0answers
377 views

The automorphism group of a field with $p^2$ elements

Suppose $K$ is a field. Then we call $f: K\to K$ a (field) automorphism if $f$ is a one-to-one, onto and unital (i.e. $f(1)=1$) homomorphism of rings. The following results are well-known. There ...
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0answers
37 views

If an $H\leq G$ has an irreducible representation of dimension $d$, then show $G$ has an irreducible representation of atleast dimension $d$. [duplicate]

Let $H$ be a subgroup of a group $G$, and let $\rho:H\rightarrow GL(V )$ be an irreducible representation of dimension $d$. Prove that there is an irreducible representation of $G$ whose dimension is ...
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0answers
47 views

Algorithm to write an element of $SL_2(\mathbb{Z})$ as a product of $S, T^n$

It is well-known that $SL_2(\mathbb{Z})$ is generated by $S = \left( \begin{array}{ccc}0 & -1 \\1 & 0 \end{array} \right), T = \left( \begin{array}{ccc}1 & 1 \\0 & 1 \end{array} ...
2
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1answer
96 views

Stabilizers of continuous profinite action on a finite discrete set

I need to prove the equivalence of categories of finite $G$-sets and the category of finite discrete set with a continuous action of the profinite completion of $G$. For any group G. I already have a ...
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3answers
54 views

Properties preserved under Homomophism

Homework question from Intro to Group Theory If $f\colon G \to H$ is a homomorphism of $G$ onto $H$, prove: If every element of $G$ is its own inverse, every element of $H$ is also it's own inverse. ...
3
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1answer
130 views

Coset Enumeration

I have read some material on Coset Enumeration. Unfortunately I could not follow the steps in Todd-Coexeter Algorithm, and also in Handbook of Computational Group Theory by Derek Holt. The problem is ...
1
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1answer
35 views

Resources on surjunctive groups.

Are there any free available resources on surjunctive groups which are available to say: a graduate level student? A textbook would be fine also. Regards.
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1answer
47 views

Locally (normal and $\mathfrak{X}$) groups II

Let $\mathfrak{X}$ be a class of group which is subgroup closed (namely $S\mathfrak{X}=\mathfrak{X}$, i.e. if $H\leq G$ and $G\in \mathfrak{X}$ then $H\in \mathfrak{X}$). Let $M\mathfrak{X}$ be the ...
2
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1answer
38 views

Question about homomorphism of groups

$G,H$ are groups, $\varphi:G\to H$ is homomorphism. How do I prove: $$\ker\varphi=\left\{{e_G}\right\} \Leftrightarrow \varphi\;\text{is injective} $$ I have problem with $\Rightarrow$ direction. ...
8
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1answer
152 views

Structure theorem for finitely generated commutative $semi$groups.

$\newcommand{\ZZ}{\mathbb{Z}}$ It is a classical theorem that a finitely generated commutative group is isomorphic to one of the form: $$ \ZZ^n \oplus \ZZ/{m_1}\ZZ \oplus \ZZ/{m_2}\ZZ \oplus \dots ...
2
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1answer
64 views

A problem about Normal Subgroup in matrices.

Let $$G=\left\{\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mid c=a^{-1},a,b \in \mathbb R,a>0\right\}$$ and $$H=\left\{\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \mid b \in ...
2
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1answer
52 views

Show that if $ \gcd(|G|,|H|) = 1 $, then $ \text{Aut}(G \times H) \cong \text{Aut}(G) \times \text{Aut}(H) $.

Let $ G $ and $ H $ be finite groups. If $ \gcd(|G|,|H|) = 1 $, then I want to show that $$ \text{Aut}(G \times H) \cong \text{Aut}(G) \times \text{Aut}(H). $$ In my attempt, I first defined the map $ ...
0
votes
1answer
67 views

Sizes of kernels of homomorphisms

I have a problem that I have been stuck on for two hours. I would like to check if I have made any progress or I am just going in circles. Problem: Let $\alpha:G \rightarrow H, \beta:H ...
3
votes
3answers
88 views

How to show that $\mathbb{Z}$ with the operation $m*n=m+(-1)^m n$ is associative?

I'm reading Beardon's Algebra and Geometry: Show that $\mathbb{Z}$ with the operation $m*n=m+(-1)^m n$, is a group. I've been thinking on how I should show associativity, I thought about doing ...
9
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1answer
171 views

On conjugacy class size of finite groups.

Suppose $G$ is a finite group such that the set of all the conjugacy class size is $\{1,2,\dots,n\}$, where $n$ is a natural number. Is it true that $n\leq 3$? Thanks in advance.
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2answers
75 views

isomorphism between two factor groups

We know that $A$ and $B$ are subgroups of $G$. Moreover, $B$ is a normal group of $G$. I've proved that if $BA=AB$ then $AB$ is a subgroup of $G$. Then I have to prove that factor group $A/(A \cap ...
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2answers
124 views

How I can prove that there is a bijection between the set $A$ and $ℤ$?

Let $f:ℝ→ℝ$ be a real analytic function and not identically zero. Assume that $f$ has infinitely many negative zeros. Let us consider the set $A$ of points $(c_{k},0)$ where $c_{k}<0$ is a zero of ...
2
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2answers
123 views

Are the groups $(\mathbb{R} \!\,, +)$ and $(\mathbb{R} \!\,^*,\dot{\,} \!\,)$ isomorphic?

I have an exercise to do and I don't understand how to solve it. It states: Determine if the groups $(\mathbb{R} \!\,, +)$ and $(\mathbb{R} \!\,^*,\cdot{\,} \!\,)$ are isomorphic and to justify ...
12
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2answers
397 views

Proof via Group Theory : $\mathrm{lcm}(a,b) \cdot \gcd(a,b) = |ab|$

Recently, I was informed that we can verify the famous formula about $\mathrm{lcm}(a,b)$ and $\gcd(a,b)$ which is $$\mathrm{lcm}(a,b)=\frac{|ab|}{\gcd(a,b)} $$ via group theory. The least common ...
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3answers
229 views

Find all possible abelian groups of order $120$.

Find all possible abelian groups of order $120$. If someone could walk me through how to do this, that would be great.
6
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1answer
82 views

On the subset $nil(G):=\{x\in G \mid \langle x,y \rangle \text{ is nilpotent for all } y \in G\}$ of a group $G$.

Let $G$ be a group and $nil(G):=\{x\in G \mid \langle x,y \rangle \text{ is nilpotent for all } y \in G\}$. Is $nil(G)$ always a subgroup of $G$? Many thanks for any help.
2
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2answers
199 views

Is the converse of Lagrange's Theorem true for the permutation group $S_5$?

Is the converse of Lagrange's Theorem true for the permutation group $S_5$? That is, if $n\mid |S_5|$, then is there a subgroup of $S_5$ with order $n$. Since $|S_5|$ = 5! = 120, then any subgroup ...
10
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3answers
735 views

Find four groups of order 20 not isomorphic to each other.

Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic. So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral ...
5
votes
0answers
35 views

Non-Isomorphic groups of order 20 [duplicate]

How many and which distinct groups of order 20 that are not isomorphic? Why are they not isomorphic? I have the $Z_{20}, Z_2+Z_{10}, D_{10}$. There are more than 4. I don't know the limit.
0
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1answer
77 views

$K \subset S_4$ contains a 3-cycle and a 4-cycle $\implies K = S_4$

In my lecture notes of algebraic number theory they use the following claim to prove that the Galois group of $X^4 +X+1$ is the whole $S_4$ If $K$ is a subgroup of $S_4$ which contains a 3-cycle ...
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1answer
61 views

Group and Subgroup Orders

If G is a finite group with fewer than 100 elements and G has subgroups of orders 10 and 25, what is the order of G? Please, explain your answer.
8
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1answer
290 views

finite subgroups of SO(3)

As is well-known, all finite subgroups of $SO(3)$, except for cyclic and dihedral groups, are isomorphic to $A_4$, $S_4$, or $S_5$. The classical proof of this fact uses the geometry of regular ...
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2answers
166 views

Prove that $PSL(2,9)\cong A_6.$

I have tried to solve the Exercise 8.12, page 227, in Joseph J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Graduate texts in Mathematics, Springer-Verlag, New York (1995). The ...
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1answer
88 views

$\pi$-separable group and its homomorphic image

Show that every subgroup and every homomorphic image of a $\pi $-separable group $G$ is also $\pi $-separable. I find a normal series of a subgroup or homomorphic image but I can not show that every ...
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2answers
940 views

Normalizers of Sylow p-subgroups

My assignment is to prove the following proposition, and I'm unsure if my proof is correct: Let $P$ be a Sylow $p$-subgroup of $G$, and let $H$ be the normalizer of $P$ in $G$. Prove that the ...
2
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2answers
119 views

The complement of a proper subgroup generates the whole group

$H$ is subgroup of $G$ with $H$ not equal $G$. Be $S=G-H$. I am being asked to prove that $\langle S \rangle=G$. Some tip to solve this? I think in $S_3$ is possible but I can´t prove.
4
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1answer
130 views

Problems on Sylow Theorems

Let $G$ be a finite group, let $p\in\mathbb{N}$ be a prime and let $$(ab)^p=a^pb^p,~~ \forall a,b\in G$$ Prove that $G$ has a unique sylow $p-$subgroup.
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3answers
134 views

Project on Simple Groups

I have to give a 15-20 minute presentation on a topic or result dealing with Simple Groups. Any ideas on what I can look into?
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2answers
113 views

Is there a way to show $\langle x,y,z: xz=zx,yz=zy,xy=yxz,x^4=y^4=z^2=1\rangle$ has order $8$?

The quaternion group has a particular presentation $$ \langle x,y,z: xz=zx,yz=zy,xy=yxz,x^4=y^4=z^2=1\rangle $$ So it must have order $8$, but can you deduce that just from the relations? The ...
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3answers
81 views

Ring theory: Ideals being equal

Question: Prove directly, without gcd computations, the following equalities of ideals. (i) $(5, 7) = (1)$ in $\Bbb{Z}$ (of course (1) = $\Bbb Z$). (ii) $(15, 9) = (3)$ in $Z$. (iii) $(X^3 −1,X^2 ...
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2answers
28 views

H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?

If G is a subgroup of GL(n;$\mathbb R$) and H = {A $\in$ G| there exists f:[0,1]$\to$G continuous, such that f(0)=A, f(1)=I}, Is H normal in G?
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1answer
65 views

(Q,+) and (C*,+) has no finite index subgroup [duplicate]

How to prove ($\mathbb Q$,+) and ($\mathbb C^*$,+) has no finite index subgroup?
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2answers
30 views

Solving a set of linear congruences

I am trying to understand group theory and conguences. I have a question on simultaneous congruences and i have a solution but i do not understand it. I want to solve the following set of ...
2
votes
2answers
390 views

Calculating the order of an element in group theory

Calculate the order of the elements a.) $(4,9)$ in $\mathbb{Z_{18}} \times \mathbb{Z_{18}}$ b.) $(8,6,4)$ in $\mathbb{Z_{18}} \times \mathbb{Z_{9}} \times \mathbb{Z_{8}} $ I have the solutions ...