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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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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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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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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19 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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24 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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$|G|=p_1p_2p_3$ distinct primes with $p_i \not | p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \not | p_j-1$ 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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20 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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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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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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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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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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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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13 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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37 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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29 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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35 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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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
32 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
28 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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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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53 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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Why do we not lose any generality by proving it only for finitely generated groups.

In the proof of following theorem, in a paper by Farkas- Here $\Delta(G) = \{ g \in G : |G:C_G(g)| < \infty \}$ and $U_1(\mathbb{Z}G) $ is the set of normalized units of the integral group ring ...
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32 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
29 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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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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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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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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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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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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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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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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43 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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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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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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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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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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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 ...
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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 ...
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2answers
45 views

Normal subgroups and index problem

Let $G$ be a finite group and $H$ and $K$ subgroups such that $H \lhd G$ or $K \lhd G$. If $gcd(|K|;|G:H|)=1$, show that $K \subset H$. I think I could prove it for the case $H \lhd G$ Let $k \in ...
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Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Prove the following:

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Note that $eR=\{er|r \in R\}$ is also a commutative ring with identity element $e$. (1) If I is an ...
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Why $c(a_1 a_2 … a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)… c(a_3))$?

If $a,b,c \in S_n$, why $c(a_1 a_2 ... a_k)c^{-1}$ is the k-cycle $(c(a_1) c(a_2)... c(a_3))$? (I need this to prove that two permutations are conjugate iff they have the same cyclic structure.)
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Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?

Let $G$ be a finite group, and $g$ be a an element of $G$. How could we go about proving $g^{|G|}=e$ without using cosets? I would admit Lagrange's theorem if a proof without talking about cosets can ...
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34 views

Proving a result concerning the size of orbits

I want to prove the following claim: Let $G$ be a finite group and $\alpha :G \to S_n$ a homomorphism. Then the size of every orbit of $\alpha(G)$ (considered as permutations on n letters) divides ...
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Prove this is a group

Prove $\mathbb Z_{235}$ is a group under multiplication. I know that a group must be closed under its operation, associative, have an identity element, and that every element must have an inverse. I ...
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30 views

Examples of Groups with only homomorphism sends every element of G to identity of H [on hold]

Give examples of non-trivial groups G and H so that the only homomorphism from G to H sends every element to the identity of H.
3
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1answer
25 views

Example of non-commuting conjugacy classes?

Let $x^G$ denote the conjugacy class of element $x$ in a group $G$ and $x^Gy^G = \{ab~:~a \in x^G, b\in y^G\}$ which contains, but may not equal, $(xy)^G$. Is there a simple example of a case where ...
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Relation between Jordan Normal Form and Irreducible Matrix Representations.

Ok so I have learned some very basic things about groups and matrix representations of groups. I have learned that it can be possible to find a "minimal basis" or "irreducible basis" for which a ...