单词

isomorphism theorems

释义

isomorphism theorems

A collection of theorems describing isomorphisms for groups (with similar versions for rings, modules, etc.) due to Emmy Noether.

First isomorphism theorem

Let f:G→H be a homomorphism of groups. Then the kernel of f is a normal subgroup of G, the image of f is a subgroup of H, and the quotient group G/kerf is isomorphic to Imf via the map gkerf ↦ f(g).

As examples:

f:z↦|z| is a homomorphism from ℂ to R with kernel S1 and image (0,∞) so that ℂ/S1 is isomorphic to (0,∞).

g:x↦lnx is a homomorphism from (0,∞) to ℂ with kernel {1} and image R so that (0,∞) is isomorphic to R.

h:x↦e^2πix is a homomorphism from R to ℂ with kernel ℤ and image S1 so that R/ ℤ is isomorphic to S1.

Second isomorphism theorem

Let G be a group, H be a subgroup of G, and N be a normal subgroup of G. Then HN = {hn : h ∈ H, n ∈ N} is a subgroup of G, and H ∩ N is a normal subgroup of H. Further, the quotient groups H/(H ∩ N) and (HN)/N are isomorphic. (This follows by applying the first theorem to the homomorphism h↦hN.)

Third isomorphism theorem

Let G be a group, K,N be normal subgroups such that N⊆K⊆G. Then K/N is a normal subgroup of G/N, and the quotient groups G/K and (G/N)/(K/N) are isomorphic. (This follows by applying the first theorem to the homomorphism g↦gN/(K/N).

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