By Randall R. Holmes

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2. 1). 4 Second Isomorphism Theorem Let G be a group and let H and N be subgroups of G with H ⊆ NG (N ) (this latter being the case if N is normal, for instance). 5, HN is a subgroup of G. It contains N as a normal subgroup so the quotient group HN/N is defined. Also, H ∩ N is a subgroup of H and it is normal (which is easily checked, although it is a consequence of the proof below), so the quotient group H/H ∩ N is defined. Theorem (Second Isomorphism Theorem). H/H ∩ N ∼ = HN/N. Proof. Define ϕ : H → HN/N by ϕ(h) = hN .

If n > 0, then Z/nZ = {0 + nZ, 1 + nZ, . . , (n − 1) + nZ} and, since these cosets are distinct, Z/nZ has order n. The theorem follows. 3 Quotient same thing as homomorphic image Let G be a group. The following theorem says that the notions “quotient of G” and “homomorphic image of G” amount to the same thing. Theorem. If G/N (N G) is a quotient of G, then G/N is isomorphic to a homomorphic image of G, namely the image under the canonical homomorphism π : G → G/N . Conversely, the image ϕ(G) of G under a homomorphism ϕ : G → G is isomorphic to a quotient of G, namely G/ ker ϕ.

More precisely, (i) G = a∈G aH, (ii) if a, b ∈ G and aH ∩ bH = ∅, then aH = bH. Proof. It is shown in set theory that any equivalence relation on a set gives rise to a partition of the set with the equivalence classes being the cells. 5. For the instructional value it provides, we give an elementary proof as well: (i) If x ∈ G, then x = xe ∈ xH ⊆ a∈G aH, so G ⊆ a∈G aH. The other inclusion is immediate. (ii) Let a, b ∈ G and assume that aH ∩ bH = ∅. Then there exists x ∈ aH ∩ bH and we have x = ah and x = bk for some h, k ∈ H.

### Algebra I by Randall R. Holmes

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