“isomorphism theorems”的概念、定义、翻译、参考文献-科学参考

isomorphism theorems

Mathematics

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).

单词	isomorphism theorems
释义	isomorphism theorems Mathematics 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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Third isomorphism theoremLet 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).

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).