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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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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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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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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21 views

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 ...
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2answers
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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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1answer
52 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 $Ker (f)$ is finite, then $G$ is finite. ...
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3answers
45 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
16 views

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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53 views

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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27 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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0answers
19 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
34 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: ...
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2answers
66 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
45 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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30 views

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
39 views

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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33 views

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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20 views

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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2answers
67 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
52 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
55 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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39 views

$|H|$ 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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47 views

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 $F$ ...
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1answer
57 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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1answer
25 views

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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63 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? $$\langle a,b:a^bb^a=(b^{-1}a^2)^2=e \rangle\cong \langle x,y:x^2=y^3=e\rangle$$
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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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0answers
21 views

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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54 views

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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39 views

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 ...
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1answer
36 views

$G$ be a group of order $p^n$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$?

Let $G$ be a group of order $p^n$ , where $p$ is a prime and $n \in\mathbb N$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$ ?
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index 2 subgroups of the infinite product of Z/2Z

Is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
2
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2answers
47 views

Permutations: Interpreting Image Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
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1answer
26 views

Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group?

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
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1answer
36 views

On the number of Sylow subgroups in Symmetric Group

If $G$ is a finite group, and $P$ is a Sylow-$p$ subgroup of $G$, then the number of Sylow-$p$ subgroups in $G$ is at most $|G|/|P|$. In the Symmetric group $S_n$, the bound is attained only for ...
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Relation between torsion in torsion free of covariant derivative and torsion free group

Is there a relationship between "torsion free" of covariant derivatives and the torsion free group? Or is this just coincidence that people use the term "torsion free" here? It is in general required ...
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1answer
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Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
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For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
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2answers
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Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
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The point of writting this isomorphism theorem like this?

In group theory there is this isomorphism theorem that doesn't seem to give any special information the way it is written. Let $T\unlhd G$ and let $S\leq G$ then $\frac{S}{S\cap T}\cong ...
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1answer
44 views

Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
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1answer
62 views

Necessity of being well-defined in Group Homomorphism?

In Group Theory, homomorphism is isomorphism when we no longer restrict to bijective map; do we still need that map to be well defined in homomorphism (like in isomorphism) or homomorphism can be ...
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How many permutations cover alternating/reverse alternating permutations?

Given integers $1$ through $2n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...