# Tour:Factsheet three (beginners)

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To summarize some of the things we have seen so far in part three:

• An intersection of subgroups is a subgroup. This is because all the three conditions for being a subgroup involve closure with respect to certain group operations, and this closure is clearly preserved on taking intersections.
• A union of two subgroups is not a subgroup unless one is contained in the other. That's because given two elements and their product, any two of them lying in a subgroup forces the third one to also lie in the subgroup.
• We can, however, talk of the subgroup generated by a union of subgroups. More generally, given any subset of a group, we can talk of the subgroup generated by that subset, in two ways: one, as all the elements expressible as products of elements in the subset and their inverses, and the other, as the intersection of all subgroups containing that subset. Thus, we can construct a notion of join of subgroups.
• For any subgroup, we can partition the whole group as a union of the left cosets of the subgroup. Analogously, we can partition the whole group as a union of right cosets of the subgroup. All the left cosets of a group look alike; they're related by left multiplication. All the right cosets also look alike, they're related by right multiplication. Further, the space of left cosets and the space of right cosets can be naturally identified with each other by the map sending every element to its inverse.
• An important application of the above general idea is Lagrange's theorem for finite groups: the order of any subgroup of a finite group divides the order of the group.