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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Prove that there is a fixed point in any subgroup $H$ of $S_4$ of order $6$.

Prove that in every subgroup $H$ of $S_4$ of order 6 there is a fixed point in {$1,2,3,4$}, i.e, there exists $1\le i\le 4$ such that $h(i)=i$ $\forall h\in H$. $Start$: Suppose there is a subgroup ...
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1answer
46 views

Does pigeonhole principle apply for all groups?

I'm reading Rosen's book and it has a proof to show that a finite subgroup (set) is closed under a composition law. It says for some $i$ and $j$, $i < j$, $a^i = a^j$ i.e, $a^i = a^i \circ ...
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1answer
105 views

class equation of order $10$

Is it a class equation of order $10$ $10=1+1+1+2+5$. As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is ...
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5answers
162 views

Number of onto and into group homomorphisms between $\mathbb Z$ and $\mathbb Z$

How many homomorphisms are there of $\mathbb Z$ onto $\mathbb Z$ $\mathbb Z$ into $\mathbb Z$ These two questions are from exercise 13, from book by John B. Fraleigh. Answer of 1. is "two ...
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24 views

Number of homomorphisms between G TO/INTO/ONTO G'

How many homomorphisms are between following groups: Type 1. $Z_{10}$ to $Z_{10}$ Type 2. $ Z$ to $Z_{10}$ Type 3. $ Z$ onto $Z $ Type 4. $ Z $ into $ Z $ Type 5. $Z$ into $Z_2 $ These type of ...
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1answer
70 views

If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$.

If $p$ and $q = 2p + 1$ are both odd primes, show that $-4$ and $2(-1)^{(1/2)(p-1)}$ are both primitive roots modulo $q$. I cannot get heads nor tails of how to even start this let alone finish ...
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1answer
54 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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3answers
58 views

The elements of $\Bbb{Z}_{20}^{\times}$

The elements of $\Bbb{Z}_{20}^{\times}$, as I understand, are all the number from 1 to 20 included that are relatively prime to 20? I am having troubles finding a coherent definition of this kind of ...
4
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1answer
132 views

Finding $\mathbf{10}\otimes \mathbf{8}\otimes \mathbf{8}\otimes \mathbf{8}$ in $SU(3)$

I know that in $SU(3)$ $$\mathbf{8}\otimes \mathbf{8} = \mathbf{27}+\mathbf{10}+\mathbf{\bar{10}}+\mathbf{8}+\mathbf{8}+\mathbf{1}. $$ How can one use this to compute $$\mathbf{10}\otimes ...
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2answers
51 views

Prove that for every $x$ in a group $G$ there is a $y$ such that $y^n=x$.

Let $G$ be a finite group and let $n$ be a natural number, relatively prime to $|G|$. Prove that for every $x$ in a group $G$ there is a $y$ such that $y^n=x$. I really need assistance when it comes ...
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3answers
118 views

On special normal subgroup of a group

Let $G$ be a group and $H$ be a subgroup of $G$ such that for any $x\in G$ we have $x^2\in H$. prove that $H$ is normal in $G$. I think this true, but can not prove it. for example this is true for ...
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1answer
79 views

Extension of Goursat's Lemma

Consider $n \geq 2$ groups $G_1,..., G_n$ each having no non-trivial abelian quotient, and let $H$ be a subgroup of $G_1 \times ... \times G_n$ such that every projection map $H \to G_i \times ...
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1answer
54 views

Is the distribution of the orders of the cyclic groups generated by elements of $S_n$ known?

A week ago I was playing around with a card-shuffle method corresponding to an element of $S_{52}$, and the order of the cyclic group generated was 272 (ie, 272 shuffles returns the deck to original ...
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1answer
25 views

How two cosets are same in this example

I was reading about cosets from example 298 in this pdf. I repeat the problem here: Let $G = \{e, a, a^2, a^3\} = <a>$ where $|a| = 4$. Let $H = \{e,a^2\} = < a^2 >$ Then ...
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15 views

Uniqueness of induced functions on reduced free groups

Let $F_n$ be the free group generated by $x_1,\cdots,x_n$ and let $K_n$ be the reduced free group, that is, $F_n$ modulo the relation that $[x_i,x_i^g]=1$ for all $i\in\{1,\cdots,n\}$,$g\in F_n$, ...
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79 views

Semigroups and solutions of equation

It is easy to prove: in a finite semigroup if for all $a$ and $b$, $ax=b$ and $ya=b$ has unique solution. then it is group. But if in a finite semigroup, if for all $a$ and $b$, $ax=b$ and $ya=b$ has ...
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1answer
54 views

Describe the subgroup $K\leq S_4$ of order 8

How do I construct the subgroup $K$ (a subgroup of $S_4$ of order $8$) ?
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1answer
31 views

one end group with positve first Betti number$\beta^{(2)}_1(G)>0$

Could anyone give me an example of a countable finitely generated (f.g.) discrete group $G$ with one end but have non-trivial $H^1(G,\ell^2G)$? To be precise, consider the following two cases. (1) ...
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63 views

Direct products of simple non-abelian groups 2

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ a simple non-abelian subgroup of $G$ such that $Z(G)\leq N$ and $\frac{G}{N}\cong A$ be a non-abelian simple group. Is it true that $G=N\times A$ ...
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45 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
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57 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
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1answer
74 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
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20 views

What is the Frobenius element (if any) of this group.

For $F = \mathbb{Q}[\sqrt{5}]$ with $p = 2$ and prime ideal over p of $ q = (2, 1 + \sqrt{5})$ with the Frobenius element defined as $$ x^{Frob_q} \equiv x^p (mod q) $$ with $Frob_q \in Gal ...
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40 views

Some subgroup of $GL_2(\mathbb{Q})$

Let's consider $GL_2(\mathbb{Q})$ and $C_2\times C_2 \times C_2$, $C_2$ - cyclic group of order 2. I can't show, that group $C_2\times C_2 \times C_2$ is not a subgroup of $GL_2(\mathbb{Q})$.I don't ...
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1answer
50 views

Cardinality of HK

Let $G$ be a group and let $H$ and $K$ be two subgroups of $G.$ If both $H$ and $K$ have $12$ elements which of following numbers cannot be cardinality of set $HK=\{hk:h \in H, k \in K\}$? 1.72 2.60 ...
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1answer
54 views

Show that $H$ is normal subgroup of $G$.

Let $H\leq G$. Show that $H$ is normal iff $xHx^{-1}=H\space \forall x\in G$. My textbook defines normal subgroup of $G$ as kernel of some homomorphism which has $G$ as domain. I showed that if $H$ ...
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1answer
54 views

Why is every coset in G a subset of G?

Suppose $G$ is a group and $H$ is a subgroup of $G$. $Ha$ is a right coset of $H$ in $G$. According to the Dover Book of Abstract Algebra p. 127, "Every coset in $G$ is a subset of $G$." I understand ...
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Generators of $ D_8$

Let G= $ D_8$ be dihedral group of symmetries of square. Find the minimal number of generators for G. My book directly writes thar answer is 2. In order to do this do we have to remember the group ...
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59 views

group homomorphisms from the real line to infinite torsion abelian groups

Hello I have question in group theory that actually originated from a question in dynamical systems. Let G be the abelian group given by the real line with addition. Let H be an infinite torsion ...
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2answers
42 views

cyclic groups -class is prime number

How can I prove that a group $G$, such that $|G| = p$, where $p \in \mathbb{P}$, is cyclic?
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58 views

Showing $G/Z(R(G))$ isomorphic to $Aut(R(G))$

I am working on this problem with lots of nesting definitions: Show that $G/Z(R(G))$ is isomorphic to a subgroup of $Aut(R(G))$. For your info, $R(G)$ is called the Radical of $G$, defined as ...
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1answer
49 views

In a group of order 21, every normal subgroup is cyclic [closed]

Let $P$ be a group of order $21$. How to prove that each normal subgroup of $P$ is cyclic?
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1answer
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proof that all subgroups of $\mathbb{Z}$ are of the form $H = b\mathbb{Z}$

In the proof that every subgroup of $\mathbb{Z}$ is of the type $H=b\mathbb{Z}$ for some integer $b$, Artin (in his book Algebra) argues that if $b\in H$, [b is the smallest positive integer in $H$] ...
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How to calculate $C_\alpha$

Let $\alpha=(1,2)(3,4,5)$, $\beta=(1,2)(3,4)$ and $\gamma=(1,2)(3,4)(5,6)$ be three permutations in $S_n,n\geq 6$. How could we calculate $C_\alpha=|C(\alpha)|$ such that $C(\alpha)$ denote the ...
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1answer
39 views

$\langle \mathbb{R}\setminus \{-1\}, *\rangle$ isomorphic to $\langle \mathbb{R}\setminus \{0\},\cdot\rangle$

Let $\langle \mathbb{R}\setminus \{-1\}, *\rangle$ be a group under the operation: $a*b= a+b+ab$. How to show that it is isomorphic to $\langle \mathbb{R}\setminus \{0\},\cdot\rangle$ where $\cdot$ is ...
2
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1answer
150 views

Online Archive of Master Thesis

I am thinking about taking the thesis route to complete my master in pure math. In anticipation of these in the coming semesters, here are my questions: (1) Do you know of any links to archive of ...
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1answer
74 views

Prove any group of order $185$ is cyclic.

This is my attempt. I am not sure as for its plausibility. $Attempt$: Let $G$ be a group of order $185$. Then $G=185=5\cdot 37$. The $Sylow-p$ subgroups are unique and normal and therefore $G$ is ...
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2answers
103 views

Does there exist an abelian group with insoluable word problem?

Does there exist an abelian group with recursively enumerable presentation and insoluble word problem? My gut says "of course not!". However, my mind keeps saying "but...doesn't $\mathbb{R}$ have ...
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1answer
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Verify my proof of $G$ is nilpotent iff $xy=yx \forall x,y\in G$ such that $(o(x),o(y))=1$.

Prove: $G$ is nilpotent iff $xy=yx \forall x,y\in G$ such that $(o(x),o(y))=1$ $G$ is finite. Is that plausible? Attempt: Suppose $G$ is nilpotnet. Then $G=P_1\times\ldots\times P_k$ where ...
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63 views

Conjugacy classes in $ D_4$

Let G be group of all symmetries of square. Find number of conjugate classes in G. I tried this question just as we do for $S_n$ that the number of conjugate classes in $S_n $ is partition number of ...
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1answer
135 views

Non Abelian group of order pq

Given primes p & q with q dividing $p-1$, construct a non-abelian group of order pq as follows: Let P have order p and let Q $\subseteq$ Aut(P) have order q. Let G $\subseteq$ Sym(P) be the set of ...
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0answers
56 views

Order of $ U(n) $

Let $U(n)$ be group under multiplication modulo $n$. For $n=248$, find number of elements in $U (n)$. As I tried to do this problem. The number of required elements are $\phi(n) $. So to calculate ...
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2answers
35 views

Cyclicity of Aut ($ Z_n $)

Let Aut (G) denote group of automorphisms of group G. Which of following is not a cyclic group.? 1. Aut$ (Z_4) $ 2. Aut$(Z_6) $ 3. Aut $(Z_8) $ 4. Aut $(Z_10) $ I know that in general Aut $(Z_n) $ is ...
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50 views

Prove this set is a group.

Show that the set $\{5, 15, 25, 35\}$ is a group under multiplication modulo $40$. Is there a relation between this and $U(8)$? I am having a really difficult time beginning this proof. All this is ...
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2answers
89 views

How do I generate group table for elliptic curves over finite fields

Can someone please explain how to generate a group table for an elliptic curve over a finite field? The number of solutions or points are about 16 and it is not possible to do them by adding each ...
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1answer
92 views

$X$ is an infinite set. Prove that $S_X$ does not have proper subgroup of finite index.

Help Denote by $S_X$ the group of permutations of $X$, i.e. the group of bijections $f:X\to X$ with composition. Do we want to show $[S_X : H] = S_X?$
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61 views

Is a matrix element of a norm continuous representation always a trigonometric polynomial?

Let $\pi:G\to {\mathcal B}(H)$ be a unitary representation of a compact group $G$ in a Hilbert space $H$. Consider a matrix element of $\pi$, i.e. a function of the form $$ ...
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33 views

Group tables for elliptic curves over primes

When constructing a group table for an elliptic curve modulo a relatively large prime $p$, say 23, are adding a few points with respect to each other enough to establish symmetry and thereby deduce ...
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2answers
36 views

Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) [duplicate]

Find Number of elements of $S_{10}$ commuting with element (1 3 5 7 9) I think we need to find order of centralizer of given permutation but how to find it?
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1answer
93 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...