# Tour:Introduction six (beginners)

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This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)

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In parts one, two and three, we introduced and analysed the notions of group, subgroup, Abelian group, coset, intersection of subgroups, join of subgroups. In part four, we looked at particular examples of groups, and applied the general notions to study these examples better. We now return to the goal of general understanding of group structure.

We'll see the following pages:

- Kernel of a homomorphism: The inverse image of the identity element under a group homomorphism.
- Normal subgroup: Defines normal subgroup, without going into the myriad properties of normal subgroups.
- Quotient map: Introduces the concept of quotient map given a group and a normal subgroup.
- Normal subgroup equals kernel of homomorphism: The kernel of any homomorphism is a normal subgroup, and every normal subgroup can be realized as the kernel of a homomorphism.
- First isomorphism theorem: A somewhat stronger version of the above statement.
- Congruence on a group: Another way of looking at quotient maps.
- Understanding the quotient map: A survey article relating the ideas of normal subgroup, quotient map, and congruence. Considers what happens in the context of more general algebraic structures, and singles out what is special for groups.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.PREVIOUS: Confidence aggregator four (beginners) |UP: Getting started (beginners) |NEXT: Isomorphism of groups