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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Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
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Why does a simple coroot $\alpha^{(i)}$ correspond to a Cartan subalgebra element $H^i$?

I read here that a simple coroot $\alpha^{(i)}$ corresponds to a Cartan subalgebra element $H^i$ and don't understand why this should be the case. Roots are the weights of the adjoint ...
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Representation theory and point groups

Hello everyone :) I have a doubt. I have the point group $C_{3v}$, which is the group $$C_{3v}= \lbrace e, C_{3}, C_{3}^{2}, \sigma_{v_{1}}, \sigma_{v_{2}}, \sigma_{v_{3}} \rbrace$$ $C_{3}$ and ...
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About the elements of Dihedral Group.

I have some difficulties finding the elements of Dihedral Group $D_8$. (Note that e.g., The order of $D_8 = 8$) I know The Geometric Approach for defining the $D_8$. But I always tend to like ...
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1answer
29 views

Central extension of the Discrete Heisenberg group $H_3(\Bbb Z)$

I want to use the Discrete Heisenberg group $(H_3(\Bbb Z),\times)$ as an example for a presentation on central extensions. $H_3(\Bbb Z) = \begin{bmatrix}1&x&z\\0&1&y\\0&0&1 ...
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Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
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2answers
28 views

Question about working in modulo?

This question is in essence asking for understanding of a step in Fermats theorem done Group style. For any field the nonzero elements form a group under field multiplication. So let us take the ...
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14 views

Why is the k-th convolution of $P_S$ is equal to ${P_S^k}$

For a random walk using transpositions on $S_n$, how can it be explained that the k-th convolution of $P_S$ is equal to ${P_S^k}$. They look to be the same intuitively but how can it written ...
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Does $\chi(g^{-1})=\overline{\chi (g)}$ hold for infinite groups

Let $\chi$ be the character of some representation $\rho:G \to GL(M)$ over $\mathbb C$. Suppose $G$ is a group, then $\forall g \in G$ of finite order $n$, $ \chi(g^{-1})=\overline{\chi (g)}$ ...
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3answers
137 views

What is a free group element that is not primitive?

A primitive element of a free group is an element of some basis of the free group. I have seen some recent papers on algorithmic problems concerning primitive elements of free groups, for example, the ...
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42 views

Lie Groups and Matrices

I vaguely remember (maybe I am making this up) this. Is there some sort of result about Lie groups (of a certain class) which classifies them as matrix Lie groups? In other words, given a Lie group G, ...
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44 views

Group Theory: How do I determine if an element generates a group?

I was asked if the group $(Z_{17} \setminus \{0\}, \cdot)$ is generated by the element $2$. I understand the concept of generating sub-groups in group theory. If I was given a group $G$ and asked to ...
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1answer
25 views

If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$ (Proof Verification)

Full question: Let H and K be subgroups of a group G. If $Ha\subseteq Kb$ for some $a,b\in G$, show that $H \subseteq K$. I constructed a proof by contradiction and I am wondering whether or not it ...
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25 views

Groups, Lagrange theorem [on hold]

True or false: If it's true I should give example Else, to prove why: Finite group with subgroup of finite index
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1answer
29 views

Special question in group theory

If we have a function $\rho:SL(2,\mathbb{Z})\rightarrow GL(d,\mathbb{C})$ where $m\in SL(2,\mathbb{Z})$ and $\mathcal{M}\in GL(d,\mathbb{C})$ and d is given, for example d=3. And we know that ...
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41 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
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24 views

Nilpotent by finite group contain characteristic subgroup of finite index

Let $G$ be nilpotent by finite group( i.e there exist normal nilpotent subgroup $H$ such that $G/H$ is finite), i want to prove that $G$ contain a nilpotent charactersitc subgroup of finite index.
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1answer
41 views

Prove that the maximal normal Abelian subgroup $A$ of metabelian group is equal to $C_G$($A$).

Let $G$ be finitely génerated metabelian group, then there existe maximal normal Abelian subgroup $A$ such that $C_G$($A$)=$A$. I want to prove that $C_G$($A$)=$A$.
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43 views

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic? [on hold]

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic?. How would you check this?
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1answer
35 views

Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
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1answer
34 views

Isomorphism of two non-abelian groups of order $pq$

Let $p$ and $q$ be two primes such that $q\mid p-1$. Suppose $\phi, \varphi$ are two non-trivial homomorphism from $\mathbb{Z}_q$ to $Aut(\mathbb{Z}_p)$. How to define an isomorphism from ...
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13 views

Weights in the Dynkin Basis and Eigenvalues of the Cartan Generators for SU(3)?

The Cartan Generators of $SU(3)$ in the three dimensional rep have eigenvalues $(1,-1,0)$ and $\frac{1}{\sqrt{3}} (1,1,-2)$. Therefore we have the weights: $$ (1,\frac{1}{\sqrt{3}}) \quad ...
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19 views

Occurrences of trivial representation is equal to dimension of $\{v\in V:\varphi(g)v=v\}$.

Suppose $\varphi\colon G\to GL(V)$ is a complex representation with character $\psi$. If $W=\{v\in V:\varphi(g)v=v,\ \forall g\in G\}$, why is $\dim W=(\psi,\chi_1)$, where $\chi_1$ is the principal ...
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39 views

Number of Sylow $p$-subgroups of a direct product of groups

Let $G$ be the group $S_4\times S_3$ . Prove or disprove the following: a $2-$Sylow subgroup of G is normal a $3-$Sylow subgroup of G is normal I've got $|S_4\times S_3|=144$ and the group as not ...
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1answer
32 views

How to complete this proof of the Orbit-Stabilizer Theorem?

Let $G$ be a group, $X$ a set, and $*$ and action of $G$ on $X.$ Let $x \in X$ and denote by $\operatorname{Orb} \left( x \right)$ the orbit of $x$ and by $\operatorname{Stab} \left( x \right)$ the ...
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37 views

Infinite non-abelian $ p $-groups.

Is it true that every nilpotent group is a solvable group? It is true for finite $ p $-groups, but I am not sure about infinite $ p $-groups.
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1answer
19 views

Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
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1answer
34 views

Show $G/N $ is cyclic

if $G$ is cyclic, and $N$ is normal to $G$, then $G/N$ is cyclic Can anyone give me a git to start this question? Thanks
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1answer
32 views

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?
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50 views

Fundamental group of projective plane with handles

I was told that the fundamental group of the projective plane with g handles is isomorphic to $\langle c_1, \ldots, c_{2g+1} | c_1^2 \cdot \ldots \cdot c_{2g+1}^2\rangle$. How can I show it? I can ...
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59 views

Is this an action of $S_{n}$ on $\mathbb{R}_{n}$?

I am trying to prove that $S_{n}$ acts on $\mathbb{R}_{n}$ with the map $$* : S_{n} \rightarrow \mathbb{R}_{n}, \quad * \left( \sigma, \left( r_{1}, r_{2}, \dots, r_{n} \right) \right) = \left( ...
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2answers
33 views

Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
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1answer
32 views

Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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1answer
36 views

Galois group of the extension $E:= \mathbb{Q}(i, \sqrt{2}, \sqrt{3}, \sqrt[4]{2})$

In order to make a smaller example for my question Galois group of the field of all constructible complex numbers, I am posing this new question. I know already, that E is a galois extension of ...
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1answer
53 views

Let $g$ be an element of the group $G$. If $|g|>1$ and $|G| = 3*5*7$ is it true that $|g|=3$, $5$ or $7$?

Let $g$ be an element of the group $G$. If $|g|>1$ and $|G| = 3*5*7$ is it true that $|g|=3$, $5$ or $7$? I think that answer is yes, because $|g| \ |\ |G|$. But on the other hand, I don't know ...
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1answer
35 views

Is it possible to bound recurrence functions for primes?

Would it be possible to bound this function for primes in terms of the maximum difference between the images of the function and their closest primes (for instance, the fifth term is 33 and has ...
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2answers
59 views

About the equation $|HK|=\frac{|H|.|K|}{|H\cap K|}$

For simplicity, consider $G$ a finite abelian group. Let $H,K$ be subgroups of $G$. Then we know that $|HK| = \frac{|H|.|K|}{|H\cap K|}$. Question: Does this relation holds if we assume that $H,K$ ...
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39 views

Show $HK \leq G$

Let $G$ be a group, let $H$ be a subgroup of $G$ and $K$ be a normal group of $G$, Show $HK \leq G$, where $HK=\{hk\vert h \in H, k\in K\}$ Proof: Since $H$ and $K$ are subgroup of $G$, ...
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3answers
53 views

Show $G$ is not abelian of order 8.

Let $G$ be a group whose elements are expression in terms of $x,y \in G$ such that $\lvert \langle x\rangle \rvert =4$, $\lvert \langle y\rangle \rvert=2$, $xy=yx^3$. Show $G$ is not abelian of ...
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Give an example of a group G with a subgroup H and a prime p such that a Sylow p-subgroup of H is not a Sylow p-subgroup of G

I don't understand why this is possible. If H is a subgroup of G then you know the order of H divides the order of G and same with P a subgroup of H so how could p not be a Sylow p-subgroup of G?
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Is it possible to find two groups $G_1$ an $G_2$ of orders seven and eight and a morphism $f: G_1→G_2$ such that $|\operatorname{Im}f|=4$? [on hold]

Is it possible to find two groups $G_1$ an $G_2$ of orders seven and eight and a morphism $f: G_1→G_2$ such that $|\operatorname{Im}f|=4$? How would you explain this?
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Assume $H\not\subset K\triangleleft G=⟨a⟩$ and a has finite order. prove $H\triangleleft G$ [on hold]

assume $H\not\subset K\triangleleft G=\langle a\rangle$ and $a$ has finite order. prove $H\triangleleft G$. I know that there is at most one subgroup of this group $G$ of every given order. But how ...
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2answers
37 views

if $P$ is a prime ideal of $O_K$, then $O_K/P$ is finite

let $P$ be a non-zero prime ideal of $O_K$, where $K$ is a number field(i.e. the degree $[K:\mathbb{Q}]$ is finite) then $O_K/P$ is finite. I'm working through a proof for this claim, however there is ...
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46 views

Let $p$ be a prime number; and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G:G]$

I have already figured out that $|Z(G)| = p$, and that $G'=[G,G] \lhd G$. Also, $|G'| = p$ or $p²$ I suppose I'd have to prove that $G' \subset Z(G)$, but I've been trying and I have no idea how. ...
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44 views

Does this theorem have a formal name?

I am looking for a referable name for this theorem if one exists. The group $Z_n \times Z_m$ is isomorphic to $Z_{nm}$ if $n, m$ are relatively prime. Thank you.
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28 views

Order of any element divides the order of the group poof

Show that the order of any element $g \in G$ divides $n$, where $n$ is the order of a group $G$. So far I have shown that: $G$ - finite group and let $g \in G$, therefore, the order of the ...
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2answers
33 views

Extension of Naturals via Grothendieck Group Construction

So there is a way of extending the set $N$ of natural numbers with 0, equipped with ordinary multiplication, to its Grothendieck group, the group of integers with respect to addition. This group, ...
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1answer
33 views

Is it true that $H = H_1 \times H_2 \dots H_r$

Suppose that $G= G_1 \times \dots G_r$ be a decomposition of group $G$ into its normal subgroups. Let $H_i \leq G_i$ for every $i$.We know that for every $i \neq j$, we have $[G_i, G_j]=1$ and so ...
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2answers
24 views

There exists a isomorphism $f$ such that for every $k \in K \,$, $f(k) k^{-1} \in H$.

Let $G=H \times K$ and $G=H \times L$ be two decomposition of group $G$ into its normal subgroups. Prove that there exists a isomorphism $f: K \longrightarrow L$ such that for every $k \in K$, we have ...
2
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2answers
71 views

Proving $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$

How do I prove $\mathbb{Q} \times \mathbb{Z_2} \ncong \mathbb{Q}$? I know that they are not isomorphic because for each element in $\mathbb{Q}$, say $\frac{a}{b}$, there are two corresponding elements ...