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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Order of an element in an external direct product

Consider $\mathbb{Z}_{4}\times \mathbb{Z}_{4}=\left \{ 0,1,2,3 \right \}\times \left \{ 0,1,2,3 \right \}$ The element $\left ( 2,0 \right )$ is of order 2 but I cannot figure out why. $2=LCM\left ...
4
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3answers
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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32 views

Center of $G_1$ x $G_2$ is $\mathbb{Z_{1}}$ x $\mathbb{Z_{2}}$

If $G_1$ and $G_2$ are groups and $\mathbb{Z_{i}}$ is in the center of $G_i$, is there a particular reason that the center of the product $G_1$ x $G_2$ is $\mathbb{Z_{1}}$ x $\mathbb{Z_{2}}$. I'm ...
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1answer
59 views

Proving Lagrange's theorem with homomorphisms

Let f:G-->H be a homomorphism, where G is a finite group with identity e1 and H is a finite group with identity e2. Prove that the order of f(g) is a divisor of g for all g in G. So I know that ...
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52 views

Show Latin Square is not a group.

If we fix the first two rows in the above figure, then there are many ways to fill in the remaining rows to obtain a Latin square. Show that none of these Latin squares is the multiplication ...
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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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1answer
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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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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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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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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83 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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2answers
262 views

Dense subgroup in a profinite completion

Let $G$ be finitely generated residually finite group and $\hat{G}$ its profinite completion. Let $H \leq \hat{G}$ be a dense subgroup. Does it follow that $\hat{H}$ is isomorphic to $\hat{G}$?
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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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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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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 ...
4
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1answer
136 views

$M(A)=\left\{g\in G:g^n=\prod_{i=1}^ma_i^{n_i},a_i\in A,n_i\in\mathbb N\cup\{0\},n\in\mathbb N\right\}?$

Let $G$ be a group and $\emptyset\neq A\subseteq G$. Let $$M(A)=\bigcap_{A\subseteq C}C$$ where $C$ is any subset of $G$ such that $c^k\in C$ for all $c\in C$ and $k\in\mathbb N\cup\{0\}$ and if ...
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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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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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3answers
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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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1answer
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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1answer
205 views

For $H \leq G$, showing that $N_G(H)/C_G(H) \leq \text{Aut}(H)$

This question probably has a very simple answer! I'm trying to understand the proof of the following result from Dummit and Foote, 3ed: Here is the proposition referenced: I don't understand ...
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1answer
70 views

Is this true: $A \le N_G(B) \not \Rightarrow B \trianglelefteq A$?

Let $G$ be a group, let $A, B$ be subgroups of $G$, and assume $A \le N_G(B)$. My question comes from reading Dummit and Foote, $\S 3.3$: The Isomorphism Theorems. We are proving the Second/Diamond ...
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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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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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2answers
551 views

a conjugate of a glide reflection by any isometry of the plane is again a glide reflection

Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ...
3
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1answer
167 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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1answer
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$G = \mathbb{Q} / \mathbb{Z}$ 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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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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Proving that $G/N$ is an abelian group

Let $G$ be the group of all $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$ where $ad \neq 0$ under matrix multiplication. Let $N=\left\{A \in G \; \colon ...
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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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2answers
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$|G|>2$ implies $G$ has non trivial automorphism

Well, this is an exercise problem from Herstein which sounds difficult: How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism? The only thing I know which connects a group ...
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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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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
33 views

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
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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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$A_4 \oplus Z_3$ has no subgroup of order 18

Here my solution: Suppose there exists and $H \leq A_4 \oplus Z_3$ such that order of H is 18. Now, notice index of H in $A_4 \oplus Z_3$ is 2. therefore, H is normal, and therefore, the $A_4 \oplus ...
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2answers
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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
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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Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements.

Let $G$ be a finite group with $|G|>2$. Prove that Aut($G$) contains at least two elements. We know that Aut($G$) contains the identity function $f: G \to G: x \mapsto x$. If $G$ is ...
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1answer
391 views

How many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads?

I am trying to figure out how many different necklaces can be make from 8 blue beads, 3 green beads, and 3 brown beads. I understand how to do the problem with two colors, but I am struggling to ...
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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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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
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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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) ...
2
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0answers
12 views

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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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?
3
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1answer
42 views