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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Isometry Group acts by isometries on space

This is a rather trivial question. But does the isometry group $\text{Isom}(X)$ of a space $X$ act on $X$ by isometries?
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38 views

On the number of conjugacy classes in $S_n$.

Let $\sigma=(1,2,11)(3,4)(5,6,7,8,9)\in S_{12}$. I am willing to get the number of conjugates of $\sigma$. Clearly if $\tau$ be one such, then it must have the same cycle type. So in other words, we ...
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1answer
33 views

$\Gamma(2)$ and $\Gamma_0(4)$ are conjugate

Show that the groups $\Gamma(2)$ and $\Gamma_0(4)$ are conjugate to each other in $SL(2,\mathbb{R})$, where $\Gamma(2)$ and $\Gamma_0(4)$ is congruence subgroups of the modular group Is it the ...
3
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0answers
60 views

The Fifteen Puzzle and $S_n$ [duplicate]

I was studying permutation groups from the book "Abstract Algebra and Applications" by Karlheinz Spindler in which page 553 I came across the following interesting problem. It is on the famous "The ...
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31 views

On conjugacy class size of finite group.

Let $G$ be a finite group and $g\in G$ such that $o(g)=pq$, where $p<q$ are prime numbers. (Also we know that $g^G$ is conjugacy class of $g$ in $G$.) Under which conditions we can say that ...
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1answer
43 views

How to obtain real irreducible representation matrices for finite point groups?

I would like to generate the irreducible representation matrices in real (not complex) form for any finite point group, in order to use them in a projection operator. At least I require the diagonal ...
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1answer
50 views

Given adjoint action find original matrix.

Given the Adjoint action of a matrix; $\text{Ad}(g) X_1 = g \, X_1 \, g^{-1} = X_2 $. Where g is in a (matrix) Lie group, $X_1,\; X_2$ are from the Lie algebra, can a $g$ be written in terms of the ...
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1answer
66 views

A few tasks from group theory

Can you tell me, if my solutions are good? Mapping $f:\mathbb{Z}\rightarrow G$ with $f(k)=g^k$ is group homomorphism and Image(f) is abelian subgroup of G with $|\langle g \rangle |=ord(g)$ ...
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2answers
43 views

Prove that $x * y = \frac{x+y}{1+xy}$ is a stable part of $G=(-1, 1)$

I have to prove that the result of $x * y \in G$ so $\frac{x+y}{1+xy} \in (-1, 1)$. So $x > -1$ and $y > -1$ at the same time $x < 1$ and $y < 1$. If I multiply the first 2 expressions I ...
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1answer
31 views

Relaxing the subgroup requirement for cosets

The definition I have for left cosets is as follows: Let $G$ be a group and let $H$ be a subgroup of $G$. A left coset of $H$ in $G$ is a set of the form $gH=\{gh:h\in H\}$ for some $g\in G$. ...
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1answer
28 views

Induced homomorphism $\phi^*$ from $G/M \to H/N\ ?$

Let $\phi : G \to H/N$ be a homomorphism where $G$ and $H$ are groups and let $M \unlhd G$ and $N \unlhd H$. Now when does $\phi$ induces a homomorphism $\phi^*$ from $G/M \to H/N\ ?$ When $M ...
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66 views

The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ ...
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8answers
2k views

Is symmetric group on natural numbers countable?

I guess it is too difficult a question to ask about the cardinality of $S_{\mathbb{N}}$ so I would like to ask whether it is countable or not. I tried to prove it is uncountable somewhat mimicking ...
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1answer
23 views

cyclic subgroups and generators

g is a generator of the cyclic group C45. Sketch the lattice of subgroups of $C_{45} $ For each subgroup, state its order and give a generator in terms of g. You do NOT need to list the ...
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1answer
20 views

Define all points in the affine integral lattice.

Define all points in the affine integral lattice $\mathcal{L}=\{(x,y,z,t) : x+y+z+t=5$ and $x-z \equiv 0$ (mod $12$)$\} \subset \mathbb{Z}^4$. This is a question from a practice exam I have with no ...
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1answer
386 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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1answer
17 views

How to create lattice diagrame in maple 14?

I am studying lattice diagrame of subgroups of groups and I have already posted one query over here. Now my present query is: I am using MAPLE 14. Can anyone suggest me how to create lattice ...
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1answer
22 views

What does $C_2^8, C_2^4$ etc in lattice diagrame of subgroup represent?

I am studying lattice diagrame of subgroups of groups. and I came to know about the lattices of $C_4\times C_2$ and $C_8\times C_2$ over here and here. But the problem is: I am unable to understand ...
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1answer
27 views

Show that the order of G = order f(G) times order ker(G).

Let $f:G \rightarrow G'$ be a homomorphism and let $H$ be the kernel of $G$. Suppose $G$ is finite. Show ord$(G)=$ord$(f(G)) \cdot $ord$(H)$. What I want to do is to construct a bijection, $\Phi$ ...
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1answer
8 views

Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
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1answer
24 views

Is there a general formula for finding all subgroups of dihedral groups?

It seems that $\{e\}, \{e,s\}, \{e,rs\}, \{e,r^2s\},...,\{e,r^{n-1}s\}, \{e,r,r^2,...,r^{n-1}\}, D_n$ are always subgroups of $D_n$. Especially when $n$ is odd, these seem to be the only subgroups. ...
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2answers
79 views

Why does $\{1 \dots 9\}$ behave like this under multiplication mod $10$?

When I multiply the set $$\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$$ by $2$ and take the remainder mod $10$, I get the following repeated pattern. $$\{2, 4, 6, 8, 0, 2, 4, 6, 8\}$$ Multiplication by any even ...
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1answer
32 views

A group of even order must contain an odd number of elements of order 2. [duplicate]

I tried it using proof by contradiction: Suppose that there are even number of elements of order $2$. Call them $x_1, ..., x_{2m}$, where $x_1,...,x_{2m} \neq e$. Then, consider the set $G=G ...
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2answers
109 views

How do we know that our definitions/axioms are not contradictory?

Let us assume that we have declared some axioms. Now, we wish to declare a new axiom too. How do we establish that the new axiom is not a consequence of, nor a contradiction to the original set of ...
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1answer
33 views

Don't quite understand the question. [Rotman: Group Theory ][Semantics question]

This is more of a "If I knew what that symbol/word/phrase meant I might understand the whole Idea" question and less of a "I am so lost I cannot even understand the question" question. Here is the ...
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1answer
24 views

List of two-sided wallpaper groups?

I'm interested in the symmetries of two-dimensional patterns that have two sides. In other words, what discrete groups can be formed from the three-dimensional Euclidean isometries which preserve a ...
4
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1answer
48 views

Induction of an irreducible group representation

I'm having some trouble finding the answer to the following question. Any ideas on how to get started? Let $H$ be a subgroup of a group $G$ and let $U_{1}$, ...,$U_{k}$ be the irreducible ...
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1answer
38 views

Question about the assumptions to have $G \simeq H\times K$

I've been looking this fact: Let $G$ be a group, with $G$ abelian. Let $H$, $K \leq G$, with $G=HK$ and $H\cap K=\{e\}$. Then, we have that $G \simeq H\times K$. And my question is: We know ...
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1answer
532 views

$ U(1) $ and $ SO(2) $ are locally equivalent.

In one of my particle physics textbooks, I came across the statement above. I don’t really know what it means. I know a bit of group theory and that $ U(1) $ is just the $ 1 $-d unitary transformation ...
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1answer
35 views

Orbit, Stabiliser help please?

Let $Q$ denote the rectangle with the vertices $C_1=(2,1), C_2=(-2,1), C_3=(-2,-1)$ and $C_4=(2,-1)$. Describe the elements of the symmetry group $G$, of $Q$. Note that $G$ permutes the edges of ...
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1answer
46 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
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29 views

Matrices within Group Theory

Recall that $GL_{2}(\mathbb{R})$ denotes the group of 2x2 invertible matrices with real entries with the product given by matrix multiplication. Let H denote the smallest subgroup of ...
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3answers
71 views

Is every normal subgroup of a finitely generated free group a normal closure of a finite set?

Let $G$ be a group, $S\subset G$ a subset, then the smallest normal subgroup of $G$ which contains $S$ is called the normal closure of $S$, and denoted by $S^G$. My question is, if $G$ is a free ...
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4answers
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Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$

Find two finite abelian groups of same order $G$ e $H$ such that $G \ncong H.$ I found groups $\mathbb{Z}_4$ e $V$ (Klein's group) that satisfy it, but would like more examples. I'm trying to use the ...
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1answer
48 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
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1answer
31 views

Geometric Representation of Quasidihedral Groups

I am going back through Dummit/Foote studying for a prelim and came across the 'quasidihedral' or 'semi-dihedral', group of order $2^n$, with presentation $\langle r,s \mid r^{2^{n-1}} = s^2 = 1, srs ...
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1answer
28 views

Why is $\operatorname{Hom}_{\mathrm{Groups}}(G,A)$ isomorphic to $\operatorname{Hom}_{\mathrm{Ab}}(G/[G,G],A)$?

This question is inspired by an exercise from the Weibel's book on Homological Algebra (beginning of chapter 6 on Group Cohomology). Let $G$ be a group and $A$ be a $G$-module. My question simply is: ...
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1answer
44 views

Looking for group of polynomials with only real roots

Assume $P_\mathbb R$ is the set of all polynomials which have only real coefficients and only real roots. Define $0$ as a polynomial with infinitely many real roots and all other constant polynomials ...
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1answer
36 views

Count number of colorings of tetrahedron, where colorings are indistinguishible if one can be reached from another by rotation

I'm fairly new to group theory, and here's one problem I'm trying to solve: We're coloring nodes of tetrahedron in 3 distinct colors, and its edges in 2 distinct colors. We're treating two colorings ...
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1answer
75 views

If $C_G(H)=N_G(H)$ for all abelian subgroups, prove that $G$ is abelian

Let $G$ be a finite group such that for all abelian subgroups $H$ of $G$, $$C_G(H)=N_G(H).$$ Prove that $G$ is abelian. ($C_G(H)$ is the centralizer, $N_G(H)$ is the normalizer of $H$ in $G$) my ...
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Help with translating from French to English

In a paper by G. Boccara, "Cycles comme produit de deux permutations de classes données," I have come across something that seems weird. On page 130, notation 1.7, it says: Si $l$ est un entier ...
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1answer
84 views

Completing Cayley table for a group

The task is to complete the following Cayley table for a given group. $e$ is of course the identity element. Together with group axioms and the fact that every Cayley table of a group must be a ...
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1answer
22 views

2-Frobenius groups of order $2^{10}.3^5.5.11$

A group $G$ is called a 2-Frobenius group if $G=ABC$, where $A$ and $AB$ are normal subgroups of $G$, $AB$ is a Frobenius group with kernel $A$ and complement $B$ and $BC$ is a Frobenius group with ...
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0answers
9 views

Partial Group with elements only comparable to 0?

The use case is for managing a digital ledger where the monetary amounts are kept private, but transactions are still verifiable through addition. To do this, I need a mathematical construct which ...
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5answers
362 views

Associativity for Magma

Say I have the operation table for a magma. I want to know whether or not the operation is associative. However, associativity is defined for an operation on 3 elements, and the operation table deals ...
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2answers
253 views

Groups of order 8 are not simple

Show that any group of order 8 is not a simple group. I know that $\mathbb{Z}_8$, $\mathbb{Z}_2\times \mathbb{Z}_4$, $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$, $Q_8$, $D_4$ are not simple. ...
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2answers
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Proving that a subgroup of a finitely generated abelian group is finitely generated

A question says: Using the isomorphism theorems or otherwise, prove that a subgroup of a finitely generated abelian group is finitely generated. I would say that for a finitely generated abelian ...
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1answer
312 views

list the distinct principal ideals in $\mathbb{ℤ}_2 \times \mathbb{ℤ}_3$

How do I find and list the distinct principal ideals in ℤ2xℤ3? I know that Z2 has 0,1 and that Z3 has 0,1,2, but I'm not sure how to list them and how to find ideals in Z2xZ3
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59 views

Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
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166 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...