For questions about the study of algebraic structures consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility.
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How to prove that $G=\langle a,b|a^3=1,b^2=1,ba=a^2b\rangle$ is the represntation of $S_3$. [duplicate]
How to prove that $G=\langle a,b|a^3=1,b^2=1,ba=a^2b\rangle$ is the represntation of $S_3$. Solution: First ,$ba=a^2b\implies a^{-2}bab^{-1}=1$. $S_3=\langle (12),(123)\rangle$. Denote $\phi: G\to S_3$... group-theory free-groups group-presentation- 1,635
Presentation of a subgroup of a given index
I am unable to find an explanation of why it is possible to compute a finite presentation of a finite index subgroup in a given finitely presented group. More particularly, if $G$ is a virtually ... group-theory algorithms combinatorial-geometry group-presentation decision-problems- 255
Show that $H \cap gK$ is either empty or is equal to the coset of $K \cap H$ in $H$ for subgroups $H,K <G$ and $g \in G$
Problem Statement Suppose $H,K < G$ are subgroups of a group $G$. Prove that for all $g \in G$, $H \cap gK$ is either empty or is equal to a coset of $K \cap H$ in $H$. A quick question to get ... abstract-algebra group-theory- 1,431
Alternating groups and linear groups
I learned from the literature that the symmetric group $S_n$ can be viewed a subgroup of permutation matrices of $GL_n(q)$ (where $q$ is a prime power), the general linear group of invertible $n\times ... group-theory permutation-cycles- 23
Show That Wigner’s Theorem Defines a Homomorphism $\operatorname{U}(2) \rightarrow \operatorname{SO}(3)$
Preliminary Knowledge We are working on the finite dimensional Hilbert space $\mathbb{C}^2$. The projective Hilbert space is given by $$\mathbb{P}(\mathbb{C}^2)=\big(\mathbb{C}^2 \backslash \{0\}\big)... abstract-algebra group-theory group-homomorphism quantum-mechanics- 232
How to make addition modular of real number
I think a question, this question just rised into my head.(I don't know but I'm feel this stupid question) In $\mathbb{Z}_n$ group, we have a properties of modular arithmetic. a , b $\in \mathbb{Z}_n$ ... group-theory discrete-mathematics modular-arithmetic ceiling-and-floor-functions- 1
Let $N\unlhd G$ and let $K$ be any subgroup of $G$ that contains $N$. Then $K\unlhd G$ iff $K/N\unlhd G/N$
In Hungerford's Abstract Algebra, they state the following Corollary: "Let $N$ be a normal subgroup of a group $G$ and let $K$ be any subgroup of $G$ that contains N. Then $K$ is normal in $G$ $\... group-theory proof-explanation abelian-groups normal-subgroups quotient-group- 509
Group generated topologically
Let $G$ be a topological group and $X,Y$ subsets such that $N = \overline{\langle X \rangle}$ and $G/N = \overline{\langle \pi(Y) \rangle}$. Show that $G=\overline{\langle X,Y \rangle}$. In that ... general-topology group-theory topological-groups- 306
If $g$ is in group $G$, then $g$ belongs to a subgroup $H$ only if $gH = H$
I have tried to prove this but I am unable to proceed. I have been successful in proving that $gH = H \implies g \in H$ but I am unable to prove the reverse. abstract-algebra group-theory- 21
Proving free group $F_2$ is not isomorphic to $F_3$ [duplicate]
I'm trying to prove the free group on 3 generators is not isomorphic to the free group on 2 generators. I have that there are many injections $F_3 \hookrightarrow F_2$ and of course $F_2 \... abstract-algebra group-theory free-groups- 204
Why $D_4$ is the biggest group generated by relations $\langle f,r | f^2 =1 ,r^4 =1 , fr=r^3f \rangle$? [duplicate]
I want to find the presentation of group $D_4= \{1, f, r,r^2,r^3, rf, r^2f, r^3f \} $. $r$ is the rotation of a square counterclockwise by 90 degree and $f$ is the action that flips the square. Here $... group-theory finite-groups group-presentation- 23
Doubt on inclusion of members in $S_3$.
Am preparing notes and faced one question as stated below, also request vetting of contents. My main question is stated in the edit below. Order of $S_n$ is given by the set of elements in it. $S_n$ ... group-theory permutations symmetric-groups- 3,623
The quotient group $(\mathbb{R}\times \mathbb{R},+)/\{(a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z}\}$
As said in the title, I'm trying to find a representation of the quotient group $(\mathbb{R}\times \mathbb{R},+)/ \{ (a+b\sqrt{2},a-b\sqrt{2}):a,b\in\mathbb{Z} \}$ by finding a homomorphism $f$ on $\... abstract-algebra group-theory representation-theory quotient-group- 696
Why a retraction from a building to an apartment is not isometric?
Let $X$ be an affine building and $\mathcal{A}$ a system of apartment. For any apartment $A\in \mathcal{A}$ and a chamber $C$ in $A$, let us consider the retraction $\rho=\rho_{A,C}\colon X\... group-theory metric-geometry coxeter-groups buildings affine-geometry- 101
Is any subgroup of finitely generated residually finite group finitely generated residually finite group (or Hopfian)? [closed]
Let $G$ be a finitely generated residually finite group. Is any subgroup of $G$ finitely generated residually finite group (or Hopfian)? How about an r-image of $G$? group-theory- 1,619
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