Cyclic group

Cyclic Group->

        A group (G,o) is called cyclic if for a∈G there are exist a∈G is in the form of aⁿ , where n is an integer. 

in mathematical form                          (a)={aⁿ :n∈Z} where a is the generator of G.

Theoram:-  

Every cyclic group is Abelian.

Proof. The elements of cyclic groups are of the form aⁿ

. Commutativity

amounts to proving that aⁿa

 = ajai.

aⁿa^m  =>a^n+m

        ^=>a^{m+n  addition of interger is commutative}

                         ^=>a^maⁿ

therefore
                  

aⁿa^m

=a^maⁿ

 ^{G is a cyclic group.}

                                                                                                              ^{hence proved}

^{Something about cyclic subgroup..}

Cyclic Subgroups

If we pick some element a from a group G then we can consider the subset of all elements of G that are powers ofa. This subset forms a subgroup of G and is called thecyclic subgroup generated by a. If forms a subgroup since it is

• Closed. If you multiply powers of a you end up with powers of a
• Has the identity. a • a^-1 = a⁰ = e
• Has inverses. The inverse of any product of a's is a similar product of a^-1 's.

But this is the long way of proving subgrouphood. Let's use our theorem that says if x and y are in the subset implies that x • y^-1 is in the subset then the subset is a group. This is simple here. If y is a power of a then so is y^-1 and so, therefore, is x • y^-1 .

A few facts about cyclic groups and cyclic subgroups:

1. Cyclic groups are Abelian.
2. All groups of prime order are cyclic.
3. The subgroup of a group G generated by a is the intersection of all subgroups of G containing a
4. All infinite cyclic groups look like the additive group of integers.

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