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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Prove that there is no permutation $\gamma$ such that $\gamma (1 2) \gamma^{-1} = (1 2 3)$

I need to prove that there is no $\gamma$ such that: $$\gamma (1 2) \gamma^{-1} = (1 2 3)$$ First of all, I'll try to write $\gamma$ in a generic way: $$\gamma = (a b c) \implies \gamma^{-1} = ...
2
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24 views

Symplectic group and Quaternion Inner product

I have a problem understanding a passage from "Naive Lie theory"(by Stillwell), here is the passage from section $3.9$ ,page $71$: The idea of treating orthogonal, unitary, and symplectic groups ...
2
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1answer
24 views

Direct Limit of finitely generated groups

Is every group the direct limit of its finitely generated subgroups? This is true for abelian groups, I have not seen this statement for nonabelian groups, so i am wondering if this is true. Seems ...
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34 views

Show that $Nil(\mathbb{Z}_n) $ is a subgroup of $\mathbb{Z}_n$ [on hold]

Show that $\mathrm{Nil}(\mathbb{Z}_n) = \{\bar{x}\in \mathbb{Z}_n\mid \bar{x}\,^m=\bar{0}\text{ for some positive integer $m$}\}$ is a subgroup of $\mathbb{Z}_n$.
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1answer
26 views

Can two representations with different dimensions be isomorphic?

For a finite group G and two irreducible representations, with different dimensions. How would I show that they can not be isomorphic?
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1answer
67 views

If $g$ is a permutation, then what does $g(12)$ mean?

In Martin Lieback's book 'A Concise Introduction to Pure Mathematics', he posts an exercise(page 177,Q5): Prove that exactly half of the $n!$ permutations in $S_n$ are even. (Hint: Show that ...
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64 views

Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...
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21 views

Proofs of Sylow Theorems [on hold]

‎let ‎‎$‎G‎$‎‏ ‎is a‎‎ ‎finite‎ group and ‎$‎‏p‎$‎ is prime. if ‎$‎P‎\in Sly‎_{p }(G)‎$‎‏‎‎ then ‎$‎O‎_{p}(G)=Core‎_{G}(P)‎$‎‏‎‎
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problem in meaning of symbol in commutator subgroup

i was reading paper "OUTER AUTOMORPHISMS IN NILPOTENT p-GROUPS OF CLASS 2, H. LlEBECK" in page 2 there is a symbol i dont get. if G is generated by a basis $a_λ, λ∈Λ$ and z∈Z⋂Φ(G) for σ(z,μ) be ...
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50 views

Permutations minus Transpositions

I want a formula that allows me to find all the permutations in $S_n$ (which is the set of all the integers from 1 to $n$) which don't contain a transposition. Attempt: Lets call $g(n)$ the ...
2
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1answer
51 views

What finitely generated amenable groups are known to be LERF?

I know that finitely generated nilpotent groups are LERF (LERF means "subgroup separated"). I'm looking for examples (many, if possible) of groups which are: Finitely generated, but infinite ...
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3answers
156 views

Why is reflection in a plane an automorphism?

I have not studied group theory, but would like to know in simple terms why reflection in a plane is an automorphism. Dr. Hermann Weyl gives the definition of automorphism in his book 'symmetry' as ...
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1answer
22 views

How is the distinction of left and right in space related to the orientation of screw?

In Dr. Hermann Weyl's book 'symmetry', he explains the difference between left and right as In space the distinction of left and right concerns the orientation of a screw. If you speak of turning ...
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1answer
30 views

Bijection from conjugacy class to the factor group by centralizer.

How different is $g^{-1}xg$ from $gxg^{-1}?$ Because proving a bijection from $g^{-1}xg$ type conjugacy class to the set of right cosets of the centralizer of $g$ in $G$ is as easy as proving it from ...
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2answers
43 views

Composition series and its number determine a group?

By Jordan-Holder thm, it is known that every finite group has a unique composition series.(Here, unique means that there is only one kinds of such series.) And it is known also that composition ...
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0answers
59 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall exclusively discuss finite commutative unital ...
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2answers
47 views

Ideals of non semi-simple group rings.

I worked for a long time on complex group rings and complex twisted group rings. In those cases the algebra is semi-simple and its structure is well understood from the decomposition to irreducible ...
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Given the basis vectors of a 10-dimensional representation of $SO(10)$, how can I compute the basis vectors of the 54-dimensional representation?

Because $10 \otimes 10 = 1_s \oplus 54_s \oplus45_a$ we can write each element of $54$ as a $10×10$ matrix. The usual basis vectors of the 10-dim rep are $$ \begin{pmatrix}1 \\0 \\ \vdots ...
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Simple example for Bilinear mapping

Notation : $\mathbb{G}$ is an additive group and $\mathbb{G}_T$ is multiplicative group of prime order $q$. Bilinear mapping $e: \mathbb{G} \times \mathbb{G} \rightarrow \mathbb{G}_T$ has to satisfy ...
3
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2answers
33 views

Elements of $S_n$ which can not be product of $\leq n-2$ transpositions

It is well known that every element of $S_n$ can be written as a product of at most $n-1$ transpositions. This theorem is proved in all the books which discuss the permutation groups. But, I find that ...
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73 views

$Ker (f)$ is finite, then $G$ is finite. [on hold]

Let $G$ be a group with identity element $e$, $f: G → G$ a homomorphism for which there is a natural $n> 1$ such that $f^n (G)$ = {e}. i. Prove that if $Ker ...
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3answers
51 views

Does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$?

I know that $G\cong G'$ and $H\cong H'$ implies $G\times H\cong G'\times H'$. But is it true for reverse? I mean, does $G\times H\cong G'\times H'$ imply $G\cong G'$ and $H\cong H'$? If so, how to ...
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1answer
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If a set $X$ contains three different elements $a,b,c$ describe $f:=t(a,b)∘t(b,c)$ and $g:=t(b,c)∘t(a,b)$. Are they equal?

The group of permutations of a set $X$ consists of all functions $f:X\to X$ that are one-to-one and onto. The group operation is the composition of functions. Of special importance are transpositions ...
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Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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29 views

Under what conditions is a ZG-module torsion-free?

If we have a ZG-module A, I was wondering if there are known condition we may imply on either A or the group G to make A torsion-free?
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28 views

Number maximum of commutators required to generate an element of the derived subgroup

Let $G$ be a group for which the center $Z(G)$ is of index $n$. How to prove that an element of the derived subgroup $G^\prime$ is the product of at most $n^3$ commutators?
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1answer
56 views

Where does the ambiguity in choosing a basis for a Lie algebra come from?

This is a follow-up to this question. For matrix Lie algebras, we can define the Lie algebra $g$ of a group $G$ as the set $T_a \in g$ that yield an element of $G$ when put into the exponential map: ...
3
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2answers
81 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
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1answer
47 views

On group with special properties

Is there a group $G$ with two the following properties:? i) $Aut(G)$ is not nilpotent, where $Aut(G)$ is the full automorphism group of $G$. ii) There exists an element $1\neq x\in G$ of order ...
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A normal subgroup of $ GL(n, K) $

Let $ F $ be a field and $ K $ be an extension of $ F $. Define the set, $$ E(n, K, F) := \{ M \in GL(n, K) , \det M \in F \} $$ Show that $ E(n, K, F) $ is a normal subgroup of $ G(n, K) $ and also ...
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Orbits of the symmetry group and the alternating group [on hold]

I have difficulties with these problems. Any solutions will be appreciated. 1) Compute the orbits of the symmetric group of the tetrahedron on the set of 6 pairs of vertices. 2) Compute the orbits ...
5
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1answer
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Recognizing action of semidirect product

I've been looking at some texts in representation theory and I see instances where the symmetric group $S_n$ and some other group, e.g., $GL(V_1) \times \ldots \times GL(V_n)$, act on a space. The ...
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Orbits of the tetrahedron [on hold]

Compute the orbits of the symmetry group of the tetrahedron on the set of $6$ pairs of vertices. What if the tetrahedron was an icosahedron with 66 pairs?
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Hypercyclic Group [on hold]

A group $G$ is Hypercyclic group if any Sylow subgroup of $G$ is cyclic. Can you please give some idea how to solve this Questions? $1)$ find example of hypercyclique group that is not cyclic. $2)$ ...
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71 views

Rubik's Cube's Group

Is there an article somewhere with an exhaustive study of the Rubik's Cube Group $G$? Such as computing some subgroups of it or exhibiting some elements of its center $Z(G)$? I tried googling it and ...
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2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
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2answers
59 views

What textbooks should I use for Trigonometry and Calculus? My basics are terrible.

I need help really bad. I have a paper coming up in two months and all topics require at least basic if not intermediate understanding in trigonometry and calculus. I don't know how I got so far - by ...
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$|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ commute.

Can anyone help me with this exercise? "Let $G$ be a group. Let $N$ be a normal subgroup and $H$ a subgroup of $G$. Assume that $|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ ...
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Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
4
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1answer
58 views

Homomorphism from $\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}\rightarrow \mathbb{Z}\oplus \mathbb{Z}$ has non-trivial kernel: elementary argument

One can give an elementary arguments (avoiding "rank") to prove that any group homomorphism $f$ from $\mathbb{Z}\oplus \mathbb{Z}$ to $\mathbb{Z}$ has non-trivial kernel: Let $f:(1,0)\mapsto a$ and ...
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44 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...
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2answers
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Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
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2answers
65 views

Automorphism of $D_6$

I need find an explicit way to express the group $Aut(D_6)$, and I have not idea how write this group, maybe this is an semidirec product of some groups but I don´t see this. thanks.
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how to apply tietze transformation? [on hold]

how to prove this isomorphism? $$\langle a,b:a^bb^a=(b^{-1}a^2)^2=e \rangle\cong \langle x,y:x^2=y^3=e\rangle$$
2
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1answer
22 views

Why is every subgroup of a finitely generated nilpotent group closed in the profinite topology?

This should be a well known claim, but what is the proof? Why is every subgroup of a finitely generated nilpotent group closed in the profinite topology?
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Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
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Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
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Any hint on : Every $A_{n}$ elemnt is $n$-cycles product. [on hold]

[Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of $A_{n}$ is ...
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Generators in group $Z^*_{p}$

show that $g=2$ is a generator of group $Z^*_{19}$ Can anyone explain me how i can show in this example and generally that an element is a generator in a group?
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1answer
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Hint to find the order of the group of $2\times 2$ matrices under multiplication [duplicate]

Let $G$ be the group of all $2\times 2$ matrices \begin{bmatrix}a&b\\c&d\end{bmatrix} where $a,b,c,d$ are integers modulo $p$ for $p$ prime such that $ad-bc\not =0$.$G$ forms a group relative ...