The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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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 $H$ ...
2
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1answer
40 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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2answers
27 views

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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3answers
38 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
30 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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0answers
36 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 ...
2
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1answer
39 views

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

Let $P$ be a group of order $21$. How to prove that each normal subgroup of $P$ is cyclic?
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1answer
38 views

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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0answers
36 views

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 ...
3
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1answer
31 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
64 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
50 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 ...
3
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2answers
75 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 ...
1
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1answer
31 views

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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2answers
40 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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0answers
25 views

Frobenius groups of order 36 [on hold]

Is there a Frobenius group of order 36? If yes, what is it's structure as semidirect product of two subgroups?
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1answer
33 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
33 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
29 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 ...
3
votes
2answers
40 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
60 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 ...
1
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1answer
73 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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0answers
42 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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0answers
28 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
30 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
20 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 ...
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2answers
20 views

Set of all distinct cosets of $\mathbb Z$ in $\mathbb R$ is not equipotent with the set of all distinct cosets of $2\mathbb Z$ in $\mathbb R$ ?

By third Isomorphism theorem , $\mathbb R/2\mathbb Z \Big/\mathbb Z/2\mathbb Z \cong \mathbb R/\mathbb Z$ ; so can we conclude that set of all distinct cosets of $\mathbb Z$ in $\mathbb R$ is not in ...
2
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2answers
51 views

Inequalities in groups.

I learnt that $(\mathbb{R},\times) < (\mathbb{C},\times)$, Which means the first is a subgroup of the second one. But in the first group inequality is defined, while it's not in the latter. This ...
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2answers
44 views

What are all the characteristic subgroups of $(\mathbb Q,+)$?

What are all the characteristic subgroups of $(\mathbb Q,+)$? As in, the subgroups $H$ such that $f(H)=H, \forall f \in \operatorname{Aut}(\mathbb Q)$? I have reduced the condition to $qH=H$, $\forall ...
4
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1answer
54 views

$G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$.

Prove $G$, $|G|=n$ is nilpotent $\iff$ $\forall m|n$, $G$ has a normal subgroup of order $m$. I got stuck in the second direction. One direction: $|G|=n=p_1^{s_1}\cdot ...\cdot p_k^{s_k}$ Where $p_i$ ...
2
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0answers
36 views

Non-isomoprhic semidirect products and their centers

I have a group $G \cong P \rtimes \mathbb{Z}_4$, where $|P| = p^2$ ($p$ is an odd prime). Do I have enough information to determine whether the two possible structures of $P$ both yield unique ...
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1answer
57 views

Does every group have a lie algebra? [on hold]

Does the complex number group {$C, *$} have a lie algebra? When does a group have a lie algebra?
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1answer
23 views

Hardness of discrete log in additive group

Why is discrete log considered to be hard in multiplicative group but not in additive group?
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1answer
25 views

Number of elements of order 2 in $ S_6$

Find number of Elements of order 2 in $ S_6$. My attempt: It is clear that permutations would be of form: (a b) (a b)(c d) (a b)(c d)(e f) Option 1. Will have $\frac{6!}{2.4!}$ permutations ...
3
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1answer
31 views

Find permutation $B$ given $B^4 =(2143567)$

Let $ B\in S_7 $ and $ B^4 =(2143567)$. Find B. How to find $B$? All I know is that $B^7 $ is identity permutation because it is a 7 cycle.so (B^4)^2 should be B?
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2answers
46 views

What is the difference between $[H, g]$ and $[h, g]$?

I am working on this problem, where $[H, g]$ is the commutator group: Let $H$ be a subgroup of $G$, show that $[H, g] = [H, \langle g \rangle]$. Before solving it, I need to understand the ...
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1answer
29 views

deduce the size of the normalizer

I'm stumped by this question: Given $$ H = \langle (1,2)(3,4),(1,3)(2,4) \rangle \subset A_5 = G$$ deduce $\lvert N_G(H)\rvert$. My attempt is to find a few elements of the normalizer and then ...
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1answer
34 views

Number of Elements of order 4 in $S_6$

Number of Elements of order 4 in $S_6$? My attempt: So we need Permutations of the form - (a b c d) Which equal $\frac{6!}{4.2!} $ And (a b c d)(e f) Which equal ...
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2answers
53 views

A general question about Isomorphism Abstract Algebra

I want to ask a general question about $p^2$-groups. How can I know if a group is isomorphic to $\mathbb{Z}_{p^2}$ or to $\mathbb{Z}_p \times \mathbb{Z}_p$ ? Thanks
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0answers
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structure of commutative algebraic groups

I've read that there is a structure theorem for commutative algebraic groups over an algebraically closed field $K$ namely they are the direct product of a semisimple group and a unipotent group. ...
0
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1answer
43 views

Finding subgroups of a group from specific order

Given the following group: $$ \left<\left\{ \begin{bmatrix}a & b \\0 & c \end{bmatrix} \mid a,b,c\in \Bbb Z_{5},a,c \neq 0 \right\} ,\:\: * \right> $$ where ∗ is multiplication. ...
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1answer
45 views

Roots of permutations [closed]

If $(10 1 7 12)(3 2 4 5 6)(11 8)(13 9) = k$ where $k∈S13$ so that $p^3=k$ and $p∈S13$, decide whether $p$ exists or not. If it doesn't, prove it. I have no idea how to start and even do this type of ...
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57 views
+50

Why a tensor product of 2x2 unitaries cannot implement a 3x3 unitary?

Let $\{v_1, \dotsc, v_m\} \in \mathbb{C}^{2^n}$ be a set of orthonormal vectors. Define a map $R_m$ from $2^n \times 2^n$ to $m \times m$ matrices as follows: $$R_m(M) := \sum_{i,j=1}^m (v_i^*M v_j) ...
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Definition of minimal presentation of a group

I'm working on a problem on the braid monodromy of complex lines arrangements in $\mathbb{C}^{2}.$ I have the following question. It's just a simple definition. However, I didn't find anywhere. Let ...
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T/F: Non-abelian group has nontrivial abelian subgroup / Nontrivial abelian group has cyclic subgroup [on hold]

Consider the following statements S: Every non-abelian group has a nontrivial abelian subgroup T: Every nontrivial abelian group has a cyclic subgroup. Then Both S and T are false S is true and ...
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2answers
54 views

Direct product of simple groups

Let $G=H_1\times H_2$, $H_1,H_2$ are simple groups. Let $L\vartriangleleft G$ ($L$ isn't trivial). Show that $L$ isomorphic to $H_1$ or $H_2$. I tried to construct "projections" of $L$ on $H_1,H_2$, ...
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0answers
44 views

Find $n$ s.t. $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$

can we determine the automorphism group of a $U$-group i.e $Aut(U(n))$ ? I need to find $n$ s.t $Aut(\mathbb Z_2 \oplus \mathbb Z_4 \oplus \mathbb Z_4 \oplus \mathbb Z_6) \cong U(n).$ ? I started by ...
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0answers
34 views

Infinite group G is polycyclic then subgroup of fitting is nilpotent.

Let G be a infinite polycyclic group i.e soluble and satisfy max. Show that the subgroup of fitting of G is nilpotent. he subgroup of fitting of G is subgroup of G generated by normal nilpotent ...
2
votes
2answers
60 views

To find posible order of $\frac {G}{Z (G)} $

If $Z (G)$ denotes centre of group $G,$ the order of quotient group $\frac {G}{Z (G)} $ cannot be $(A) 4.$ $(B)6.$ $(C)15.$ $(D)25.$ I know that if $G/Z(G)$ is cyclic then $G$ is abelian.
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0answers
34 views

$U(n) \simeq \frac{SU(n) \times U(1)}{\mathbb{Z}_{n}}$ isomorphism

I'm trying to proof the following isomorphism $$U(n) \simeq \frac{SU(n) \times U(1)}{\mathbb{Z}_{n}}$$ So I'm using the first Isomorphism theorem: http://en.wikipedia.org/wiki/Isomorphism_theorem ...