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by Papapetros Vaggelis
Tue Nov 10, 2015 12:07 pm
Forum: Algebraic Structures
Topic: On group theory
Replies: 2
Views: 2851

Re: On group theory

Thank you mr.Demetres. Here is a more analytical solution about \(\displaystyle{\left(\rm{Aut}(V_{4}),\circ\right)\simeq \left(S_{3},\circ\right)}\) where \(\displaystyle{V_{4}=C_{2}\times C_{2}=\left\{e,a,b,c\right\}}\) is \(\displaystyle{\rm{Klein's}}\) group. Let \(\displaystyle{X=\left\{a,b,c\ri...
by Papapetros Vaggelis
Mon Nov 09, 2015 3:38 pm
Forum: Algebraic Structures
Topic: On group theory 4
Replies: 1
Views: 2733

On group theory 4

Find all the functions \(\displaystyle{f:\left(\mathbb{Z}_{6},+\right)\longrightarrow \left(S_{3},\circ\right)}\) having the property \(\displaystyle{f(x+y)=f(x)\circ f(y)\,,\forall\,x\,,y\in\mathbb{Z}_{6}\,\,\,\,\,\,}\) (homomorphism) .
by Papapetros Vaggelis
Mon Nov 09, 2015 3:37 pm
Forum: Algebraic Structures
Topic: On group theory 3
Replies: 1
Views: 2070

On group theory 3

Let \(\displaystyle{\left(G,\cdot\right)}\) be a group and \(\displaystyle{N}\) a normal subgroup of \(\displaystyle{\left(G,\cdot\right)}\), that is \(\displaystyle{N\trianglelefteq G}\) . Prove that there exists a subgroup \(\displaystyle{K}\) of the group \(\displaystyle{Z(G/N)}\) such that: \(\d...
by Papapetros Vaggelis
Mon Nov 09, 2015 3:36 pm
Forum: Algebraic Structures
Topic: On group theory 2
Replies: 2
Views: 2907

On group theory 2

Find all the non-isomorphic abelian groups of order \(\displaystyle{300}\) .
by Papapetros Vaggelis
Mon Nov 09, 2015 1:58 pm
Forum: Algebraic Structures
Topic: On group theory
Replies: 2
Views: 2851

On group theory

Let \(\displaystyle{\left(G,\cdot\right)}\) be a group such that the group \(\displaystyle{\left(\rm{Aut}(G),\circ\right)}\) is cyclic.

Prove that the group \(\displaystyle{\left(G,\cdot\right)}\) is an abelian group.

Is the converse true ?