Normal Subgroups and Quotient Groups
Cosets can be multiplied consistently exactly when the subgroup is normal. The quotient group that results turns the kernel of a homomorphism into a complete description of the map.
What a learner can do afterwards
- Decide whether a subgroup is normal, by conjugation or by its index
- Write the multiplication table of a small quotient group
- Use the first isomorphism theorem to identify the image of a homomorphism
1 · Read
You test normality by conjugation: a subgroup is normal when conjugating any of its elements by any group element stays inside it. You may skip the work in two cases, since every subgroup of an abelian group is normal and every subgroup of index 2 is normal. For a failure, conjugate the swap of 1 and 2 by the swap of 1 and 3 to get the swap of 2 and 3, which escapes.
You find quotient sizes by dividing: a group of order 6 by a subgroup of order 2 leaves a quotient of order 3. The even residues modulo 6 are 0, 2, and 4, so that subgroup has 3 elements and yields 2 cosets, the evens and the odds. The symmetries of the square number 8 with a center of 2, so that quotient has order 4.
You read every homomorphism through its kernel with the first isomorphism theorem: the image looks exactly like the domain modulo the kernel. A map from the cyclic group of order 6 onto the cyclic group of order 3 has kernel of order 2, and its image is the whole codomain. The sign map on permutations of 3 letters hits both 1 and minus 1, so its image is that pair.
You should check normality before you multiply cosets, because the product is consistent exactly when left and right cosets agree. In an abelian group you may skip the test, since every subgroup there is normal. Never write a quotient table until the check passes.
Normality makes coset multiplication consistent, quotients divide orders, and every image is a quotient by its kernel.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.