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

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What does the Notation of $G_1 \oplus G_2 \oplus \ldots \oplus G_n$ Typically Indicate?

In the context of group theory, what does does the notation of $$ G = G_1 \oplus G_2 \oplus \ldots \oplus G_n $$ typically denote? I'm assuming its distinct from the external direct product of ...
2
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1answer
55 views

Representation of $GL_2$ on $K^2$

In one of my problems it says the following: Let $K$ be an infinite field. Consider the linear action of $GL_2$ on $K[x,y]$ induced by the natural representation of $GL_2$ on $K^2$. I don't know what ...
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1answer
42 views

Conjugacy Class Proof

I don't follow the answer to b). I understand the first couple of sentences, as they involve a lemma from the notes but i am lost thereafter. ...
1
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1answer
61 views

Sylow theorem application

Suppose that G is a group of order $60$ and $G$ has a normal subgroup $N$ of order $2$. Show that $G$ has normal subgroups of order $6$ and $10$ no of $3$ Sylow subgroups are 1 ,4 or 10 (by Sylow's ...
4
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1answer
27 views

Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
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0answers
17 views

How a surjective group homomorphism induces chain map on complexes?

I have following confusion while reading a statement: Let $G$ and $H$ be two groups and $\phi : G \rightarrow H$ be a surjective homomorphism. Let $B(G)$ and $B(Q)$ denote the bar resolutions of $G$ ...
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2answers
28 views

Is a cartesian product a group?

Let $J= N \times R$ with operation $(a,x)*(b,y) = (a+b, (ax+by)/a+b)$. Is $(J,*)$ a group? It is a cartesian product between natural numbers and real numbers. I am not sure how to prove whether it is ...
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1answer
29 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
2
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1answer
17 views

Subgroup of order $p$ is normal

I am trying to show that an arbitrary group $G$ of order $p^n$ has a normal subgroup of order $p$. My first instinct is to say that by Cauchy's Theorem, there is some element $x \in G$ such that the ...
2
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1answer
36 views

Matrix linear algebra generators

Linear algebra and special-linear group experts please help: It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here $$ C=\begin{pmatrix} 0& -1& ...
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1answer
39 views

Is there a way to encode a ring into a group?

Is there a meaningful bijection between tne set of all rings and the set of all groups? Thanks.
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44 views

Conjugacy classes

I don't seem to be able to follow this part of the proof. Why are we able to say $x \sim y$? Why does it suffice to prove that $y \in H$? Let $G$ be group. $(a)$ We say that elements $x,y \in ...
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60 views

Problem of Galois Extension

$\Bbb K$ is a non-Galois extension of $\Bbb Q$ and $[K:\Bbb Q]=4$. If $\Bbb F$ is the Galois closure of $\Bbb K$ then show that $Gal(\Bbb F/\Bbb Q)$ is either $S_4, A_4$ or $D_8$ with order 8. ...
0
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1answer
51 views

Subgroups of direct products

Consider a group $G$ which is a direct product of two groups of coprime order: $G = G_1 \times G_2$ with $|G_1|=n_1$, $|G_2|=n_2$ and $\textrm{gcd}(n_1, n_2)=1$. Let $H \le G$. Is it true that ...
2
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1answer
60 views

When is $\langle x,y\rangle$ equal to $\langle x\rangle\langle xy\rangle$?

I would like to know necessary and sufficient conditions on $x$ and $y$ to have $\langle x,y\rangle=\langle x\rangle\langle xy\rangle$. Sure that: $\bullet$ $\langle x\rangle$ and $\langle ...
3
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2answers
51 views

Galois Theory problem (primitive roots of unity)

If $e_1,e_2......,e_{p-1}$ denote primitive $p^{th}$ roots of unity. Here $p$ is prime. And set the sum of the $n^{th}$ powers of the $e_i$ as $p_n=e^n_1+e^n_2......+e^n_{p-1}$ Now I want to show that ...
3
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1answer
36 views

let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ .

Let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a sub group of $S_4$ of order $6$ . Show that $\exists~ i \in \{1,2,3,4\}$ which is fixed by each element of $H$. Attempt: ...
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4answers
42 views

Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$

Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if ...
3
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0answers
38 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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3answers
64 views

The implication sign of Group Closure

I know that $x, y \in G$ implies that $xy\in G$, but does the implication go the other way as well?
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3answers
35 views

Let $K$ be a Sylow subgroup of a finite group $G$. Prove that if $x \in N(K)$ and the order of $x$ is a power of $p$, then $ x \in K$.

Let $K$ be a Sylow subgroup of a finite group $G$. Prove that if $x \in N(K)$ and the order of $x$ is a power of $p$, then $ x \in K$. This is how I tried... Since $K$ is normal in $N(K)$, the ...
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2answers
39 views

Find $b \in G$ for each $a \in G$, such that $b^m = a$, where $m$ is coprime to the order of group G

Let $(G, \cdot)$ be a finite group (with identity element $1 \in G$) and $m \in \mathbb{N}$ be coprime to $|G|$. Proofe that for all $a \in G$ there exists exactly one $b \in G$, such that $b^m = a$. ...
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2answers
33 views

Abstract Algebra: If $H$ and $K$ are subgroups of an abelian group $G$ of order $m$ and $n$, prove $G$ has a subgroup of order $\mathrm{lcm}(m,n)$

Let $G$ be an abelian group and $H$ a subgroup of order $m$ and $K$ a subgroup of order $n$. Prove that G has a subgroup of order $\mathrm{lcm} (m,n)$ (lcm = least common multiple). any thoughts? ...
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0answers
40 views

double coset represntatives

Let $H$ be a subgroup of finite index in the group $G$. Let $g\in G$. We use $r\in HgH/H$ as notation for $g\in R$, where $R$ is a complete set of representatives for $HgH/H.$ Proof or disproof: If ...
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1answer
33 views

Coset,sylow theorems

If $|G|=n$ and $p^\alpha | n$ then $ \exists H\leq G$ with $|H|=p^\alpha$. I got stuck in one part of the proof, can anyone clarify? (induction on $n$) Suppose for inductive hypothesis that ^ holds ...
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24 views

Find point of maximal order $\operatorname{lcm}(a,b)$ on a curve

In Elliptic curve okamoto uchiyama there is a condition for picking base point $G$ such that $G$ belongs to $E_n$ of maximal order $\operatorname{lcm}(|E_{p2}|,|E_q|)$. I am not getting these ...
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33 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
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42 views

Todd-Coxeter algorithm: coincidences

I'm trying to understand the Todd-Coxeter algorithm with the help of a multiplication and relator table, but there is one thing about coincidences that is not really clear. For some small groups (for ...
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1answer
36 views

How to compute cyclic notation (23)(1)

I seem to become confused whence computing simple cyclic notations as such. From my understanding, the rule goes by starting from the right and to the left. However by doing this I only end up with ...
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2answers
137 views

Why free presentations ? Why not permutation or matrix representations?

Two days ago, I asked why free presentations? and frankly I did not get a convincing answer. I am trying here to ask the question in a different way : We know that a group can be defined by a ...
3
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1answer
51 views

3-Dimensional analogue of a Dihedral Group

I'm reading about dihedral groups right now, and I'm being asked to find the orders of various groups of rigid motions of 3-dimensional polygons. What I want to know, though, is if there's a nice ...
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1answer
35 views

Show that $T(X) =\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite.

If $S$ is a nonempty set and $X\subset S$, Show that $T(X)=\{f\in A(S)|f(X)\subset X\}$ is a subgroup of $A(S)$ if X is finite. Note: $A(S)$ is called "symmetric group". It's actually a collection of ...
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1answer
33 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
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2answers
78 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
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2answers
40 views

Group Homomorphisms and Kernels

Consider the group $\mathbb{Z}^+$. Define $f : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ by $f(x,y) = x+y$. Show that $f$ is a homomorphism and find the kernel. Detirmine whether the kernel is a ...
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3answers
65 views

Can you prove this..?

If $H$ is a normal subgroup of a group $G$ and $g^{2}\in H$ for all $g\in G$ then $[G:H]\leq 2$. I have no idea about this. As we know $H\unlhd G$ $\frac{|G|}{|H|}=2$. We can claim the result and ...
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34 views

Prove that Q/Z contains elements of every possible finite order.

I know how to prove every elements of Q/Z has finite order, since b(Z+a/b)=Z but how about proving that Q/Z contains elements of every possible finite order?
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1answer
25 views

General Linear Group is(not) compactly generated

We know that any connected Lie Group is compactly generated. I have a feeling that $\mathbb{GL}_n\mathbb{R}$ is not compactly generated. Is it true? If it is how can I prove this? If not, what ...
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2answers
27 views

The Direct Product of Groups and their Subgroups

Let $G_1$, $G_2$ be groups of prime power order. Write $|G_1|=p^m$ and $|G_2|=q^n$ for some $0 \leq m,n$. (The primes $p,q$ need not be distinct.) Let $H_1$ be a subgroup of $G_1$ and let $H_2$ ...
3
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1answer
58 views

Problem about Galois Extension.

$Gal(\Bbb K/\Bbb F)\cong S$. Where $\Bbb K$ is extension of $\Bbb F$. And $S= G_1\times G_2\times\cdots\times G_K$ is a solvable group. Where each $G_i$ are groups of prime power order and $o(S)= n.$ ...
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1answer
39 views

A question on solvable groups.

Let's suppose $S$ is a group and can be written as $S= G_1\times G_2\times\cdots\times G_K$; here each $G_i$ is a group of prime power order. And $o(S)= n.$ Is this $S$ solvable group? If yes how will ...
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0answers
30 views

Characterization of subgroup dual to Frattini Subgroup

Let $G$ be a group and let $\mathcal{L}(G)$ denote the complete lattice of subgroups of $G$. We have that every automorphism of $G$ induces a lattice-automorphism on $\mathcal{L}(G)$. From here we see ...
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1answer
35 views

Construct a group isomorphism

Construct a group isomorphism $\phi : G_1 \to G_2$, with $G_1 = U(20)$ and $G_2 = \mathbb Z_2 \oplus \mathbb Z_4$. EDIT: removed mapping because it was not an isomorphism
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2answers
28 views

How many elements does $\mathbb F = \mathbb Z_{7}[x]/I$ contain?

Let $p(x) \in \mathbb Z_{7}[x]$, given by $p(x) = x^{2}+3x+1$ and let $I = <p(x)>$ be the ideal in $\mathbb Z_{7}[x]$ constructed by $p(x)$. How many elements does $\mathbb F = \mathbb ...
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2answers
57 views

Is Dihedral group just for small order?

I'm sorry that the title is not really specifying a question, but i cannot think of a single sentence describing this question. Dihedral group is defined to be rotations and reflections of $n$-agon ...
4
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1answer
119 views

Subgroups of abelian $p$-groups

Let $A$ be an Abelian group of prime power order. It can be expressed as a (unique) direct product of cyclic groups of prime power order: $A = \mathbb{Z}_{p^{n_1}} \times \cdots \times ...
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33 views

Iterated commutator generating by iterated commutators

Let $G$ be a group and let us define by induction $G_0=G$ and $G_{k+1}=[G_k,G]$ for all $k \geq 0$. If $S \subset G$, an iterated commutator of length $n$ over $S$ is an element of $G$ of the form $$ ...
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3answers
60 views

The meaning of $\bigcap_{x\in G} x^{-1}Hx$ and the proof for the fact that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$

If $H$ is a subgroup of $G$, let $N=\bigcap_{x\in G}x^{-1}Hx$. Prove that $N$ is a subgroup of $G$ such that $y^{-1}Ny=N$ for every $y\in G$. What does $N=\bigcap_{x\in G}x^{-1}Hx$ mean? I'm confused ...
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2answers
51 views

Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.

If $A$ and $B$ are finite subgroups, of orders $m$ and $n$, respectively, of the abelian group $G$, Prove that $AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime. Definition: ...
2
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1answer
40 views

A group acts on a disk

Let $G$ be a group, it acts continuously on a disk $\mathbb{D}$, $g$ is a non-trivial central element of $G$. The set of fixed points of $g$ is $\partial\mathbb{D}$, I want to prove that for $h\in G$, ...