Subgroups Part 3

A subgroup is a subset of a group that with the restricted composition table inherited from the parent group satisfies the axioms of group theory as follows:
The subgroup’s restricted composition table must be closed and only contain elements of the subgroup.
The identity must be within the subgroup.
Associativity will be effortlessly inherited from the parent group.
For all elements within the subgroup, their inverse must also be in the subgroup.
In this video we explain this definition in detail and then go through some examples.
One of the examples we look at is the subgroups of the group of integers under addition. We find all the possible subgroups of this group and show how all but the trivial subgroup are isomorphic to the original group.

[@Gismho]

Superb explanation. Very easy to comprehend. Repetition of all statements appreciated to gain proper understanding of derivations. Many thanks.

[@changdagong3305]

Very amazing video!!!!

[@louismallet6724]

Wait ! a is not a set here, it is in the set...

[@larshizzleramnizzle3748]

Correct!

[@heleen9841]

I'm confused by the statement a = {0,1,2,....}

[@shadaanhassan5936]

​@@heleen9841 no need to confuse haleen.. here a is any number which you can pick up from the set {0,1,2,3,.....} Don't mix this set with set of integers Z. Suppose you choose a to be 3 . Then 3 is multiplied with set Z that is 3Z. further which you get know that is a subgroup of Z . As it follows all the axioms of the group Theory and 3Z is a non empty subset of Z as well. Hope your doubt is cleared.

[@xinhaofan396]

Too much repeating.

[@mohamedakrout9742]

I guess he is trying to make the videos independent of each other as much as possible.

[@DaiMoscv]

you could always skip around it's a prerecorded video after all.

[@michelef406]

"Repetita Iuvant"