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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$x$ in group G with order $r$, $y$ in group $G'$ with order $s$ what is the order of $(x,y)$ in $G$ x $G'$

I have an element $x$ of order $r$ in a group $G$ and an element $y$ in group $G'$ of order $s$. Is the order of $(x,y)$ in the product group $G$ x $G'$ $lcm(r,s)$? Thoughts: I think that this is ...
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39 views

Given $P,Q$ with prime order, prove $P \cap Q$ is trivial group?

Suppose $P,Q \leq G$ both have prime order, with $P \neq Q$. Prove that $P \cap Q$ is the trivial group. I think Sylow's theorem applies here but I feel like there is not enough information to ...
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1answer
23 views

An algorithm to find a subgroup generated by a subset of a finite group

I'm currently writing a library on python, and now I'm a little bit stuck on how to find a subgroup generated by a subset $S$ of the group $G$. In the case $S = \{a\}\subseteq G$ the problem's easy: ...
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34 views

Possibilities for a group $G$ that acts faithfully on a set of objects with two orbits?

A group $G$ acts faithfully on a set $X$ of 5 objects. The action has two orbits: one of size 2, and one of size 3. What are the possibilities for the group $G$? I think I should apply the ...
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54 views

Upto which prime is the calculation of $gnu(2^4\cdot3^4\cdot…\cdot p^4)$ feasible?

The determination of $gnu(n)$, the number of groups of order $n$ upto isomorphism, is very hard in general. But if no large powers are involved, it should be possible for relatively large numbers. ...
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1answer
69 views

Automorphisms of infinite abelian groups

It is well-known that the map $Aut$ from the class of groups to itself has fixed points. For $n \neq 2$ or $6$, $Aut(S_n) \cong S_n$, $Aut(D_4) \cong D_4$ and if $G$ is a finite non-abelian simple ...
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133 views

If $G$ is a finite group and $H \subset G$ is closed, must $H$ be a subgroup?

Supposed theorem (from online notes): If $(G,*)$ is a finite group and $H\subset G$, $H$ is non-empty and $H$ is closed under $*$ then $(H,*)$ is a group. The proof given is a real mess, but, after ...
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82 views

What is the one-one and onto mapping?

I'm reading the following paper, and here I can't understand the first step of the algorithm. Please explain what is the one-one and onto mapping? And how he gets these tables ? Commutative ...
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2answers
121 views

Application of first isomorphism theorem

Let $G$ and $K$ be groups and let $G\times K$ be the direct product of these two groups. Find a normal subgroup $N$ such that $(G\times K)/N\cong G.$ I think I need to use the first isomorphisms ...
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1answer
22 views

Show that $N \lhd G \times H \not \Rightarrow N = N_1 \times N_2$ with $N_1 \lhd G$ and $N_2 \lhd H$

I'm trying to prove the following assertion: Show that $N \lhd G\times H \not \Rightarrow N = N_1\times N_2$ with $N_1 \lhd G$ and $N_2 \lhd H$ What I tried to do is find $(n_1;n_2) \in N$ such that ...
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39 views

finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6

If I have a finite group $G$ that has an element $x$ of order 10 and another element $y$ of order 6, is there anything special about G that we can infer? Would the order of $G$ be 30?
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1answer
38 views

A problem in group theory_dsom [closed]

Let $H$ be a group of integers modp, under addition, where $p$ is a prime number. Suppose that $n$ is an integer satisfying $1 \leq n \leq p$, and let G be the group $ H \times H \times \cdots \times ...
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24 views

Show that There exists a canonical injective homomorphism between $G$ and $G\times H$

Let $(G; *)$ and $(H; \cdot)$ be two groups. The product $G\times H$ is defined by: $G\times H:= \{(g;h)\mid \forall g \in G\text{ and }\forall h \in H\}$ Show that that there exist a canonical ...
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1answer
25 views

Order in quotient group $G/H$ is not the same in $G$?

$H$ is a normal subgroup of $G$ and $p$ is prime, then $$ord_{G/H}(gH) = p \Rightarrow \exists m \in \mathbb{N}\backslash\{0\}: ord_G(g) = mp$$ Can someone explain why $ord_G(g)$ isn't just $p$?
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16 views

Hall subgroup containing all normal $\pi$-subgroups

Let $G$ be a finite group. If $H\leq G$ is a Hall $\pi$-subgroup, show that $H$ contains every normal $\pi$-subgroup of $G$. This is question is proved in some notes of mine. It starts of by letting ...
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61 views

Why do the characters of an abelian group form a group?

I was reading through Serre's Linear Representation Theory book and encountered a question to show that the set of all irreducible characters of an abelian group form a group. The proof of closure ...
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3answers
56 views

How can I show that $G$ is non abelian of order 20?

Problem says: Let $G=\langle x,y|x^4=y^5=1,x^{-1}yx=y^2\rangle$. Show that $G$ is nonabelian group of order 20. To show it, I tried to turn $x^ny^m$ into $y^kx^l$ for some $k,l$. Since I have ...
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33 views

If G contains a normal subgroup $N \cong \mathbb{Z}_2$ and $G/N \cong \mathbb{Z}$, then $G\cong \mathbb{Z}\times \mathbb{Z_2}$.

If G contains a normal subgroup $N \cong \mathbb{Z}_2$ and $G/N \cong \mathbb{Z}$, then $G\cong \mathbb{Z}\times \mathbb{Z_2}$. I'm trying to create an isomorphism $\phi : G \rightarrow ...
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21 views

Motivation for proving any nontrivial normal subgroup of $A_5$ has a 3-cycle?

This question is taken from M.A. Armstrong's book Groups & Symmetry. Question 15.12. This aim of this problem is to introduce the concept of a simple group. It asks us to first work out the ...
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1answer
27 views

Let $H,K$ be subgroups of $G$. Show: $H \trianglelefteq K \Rightarrow K\subset N_G(H)$.

Let $H,K\leq G$. Show that: $$H\trianglelefteq K\Rightarrow K\subset N_G(H).$$ How can I show that a subgroup is normal of a subgroup? For the last part, I've found a proof.
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28 views

Order of Group of 2*2 matrix [duplicate]

Let G be the group of 2*2 matrices [ a b ; c d] where a,b,c,d are integers modulo p, p is prime number, such that ad-bc≠0. G forms group under relative to matrix multiplication. What is o(G)? Let H ...
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34 views

Is the following function a homomorphism?

Is the following function a group homomorphism? $f:G\to G'$, where $G=(\Bbb{R},*)$ and $G'=(\Bbb{R}^+,o)$, and $f(x)=e^x$.
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1answer
166 views

What exactly is a p'-group?

In the context of finite group theory I understand that $p$-group is a group whose order is a power of $p$ ($p$ a prime number) but I am unclear on the exact definition of a $p'$-group. This ...
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57 views

Prove that a free group of rank $\ge2$ is centerless and torsion-free

This exercise is from Rotman's Introduction to the Theory of Groups. It's just as the title states: prove a free group of rank $\ge2$ is centerless torsion-free. Here, the definition of a free group ...
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1answer
33 views

$G = \mathbb{Q} / \mathbb{Z}$ surjective map and kernel isomorphism

Let $G = \mathbb{Q} / \mathbb{Z}$, written additively. For all $n > 0$ how come $p_n(x) = nx$ is a surjective homomorphism from $G \rightarrow G$ and how come the kernel of $p_n(x)$ is isomorphic ...
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40 views

Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
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1answer
22 views

Every subgroup of finite index contained in an infinite group $G$ contains a normal subgroup of $G$. [duplicate]

Let $H<G$ such that $H$ has finite index in the infinite group $G$. Then $G$ contains a normal subgroup of finite index contained in $H$. Can I create a subgroup of index $2$ in $G$ using elements ...
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24 views

Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
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1answer
20 views

Image and Kernel of composition of two homomorphisms

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there ...
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27 views

Question about normal subgroups and conjugacy

Is the following true? I would prefer if a hint can be provided rather than a full solution. Let H be a subgroup of the group G. If, for a fixed $g \in G\setminus H$ and a fixed $h_1 \in ...
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1answer
51 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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SU(N) tensor product decomposition

Let's consider the group SU(N). The adjoint representation is $\textbf{Adj}= $ $\textbf{N}^2\textbf{-1}$. The following decomposition holds generally ( have a look at this ref ) $$ ...
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Condition for appearance of singlet in product of two irreps.

By inspecting tables for tensor products of two finite-dimensional irreps of common Lie groups, I've noticed that a trivial subrepresentation only appears when the two irreps are conjugate of ...
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1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
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1answer
59 views

Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here ...
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Automorphisms of group von Neumann algebras

I study group von Neumann algebras $L(G)$, and I extremely want to know about automorophism (groups) of these algebras. Is there any good reference about this? I appreciate of everybody help.
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51 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
3
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1answer
46 views

Is the following equation equivalent to the 2-cocycle condition?

Given a finite abelian group $G$, I'm looking for functions $\rho:G \times G \to U(1)$ such that 1) $~~~~~\rho(g,e) = 1 = \rho(e,g)$, where $e\in G$ is the unit element and such that 2) ...
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36 views

Difference between conjugacy classes and subgroups?

I am studying Group theory and Im not sure I understand the difference between a conjugacy class and subgroup?
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41 views

Trancendental extension Galois group

Let $K$ be a field and consider the extension $K(X)$ of rational functions with coefficients in $K$. It is common knowledge that $\text{Gal}(K(X)/K)$ is isomorphic to the group ${PL }_2(K)$, which is ...
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1answer
48 views

Group generated by $x , y$ is non-commutative when $x^2 \cdot y^{-3} = I$.

The problem: Suppose group $G$ with generators $x$ and $y$ is defined by the relation $x^2 \cdot y^{-3} = I$. It is necessary to show that the group is non-commutative. I failed to solve the ...
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1answer
65 views

Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
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Linear represenation of a group(can be infinite also)

Let G be a group and let $\sigma :G \rightarrow GL(V) $ be a representation of G. Assume $\sigma$ is reducible. That is $\sigma=\sigma_1 \oplus \sigma_2\oplus .... \oplus \sigma_k $ or interms of G ...
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1answer
57 views

Any nonabelian group of order 12 is isomorphic to A4, D6, or Z3 X Z4 [closed]

Can someone show me the proof for : Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$ Ive seen a few proofs where this is included in also ...
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1answer
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A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
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1answer
46 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow ...
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1answer
79 views

Is there any efficient algorithm for computing all semigroups of order n?

I found a paper, but here the author solves a bit different problem. My question is: Is there any efficient algorithm for computing all semigroups of order n?
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1answer
54 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
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1answer
33 views

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$ [closed]

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$. Please, I appreciate any help, since I have some ideas, but those are ...
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54 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...