I understand the first isomorphic theorem, which states that a homomorphic image of a group is isomorphic to the quotient group formed by the group G and the kernel of group G.
1. Is this theorem only true for Kernel K, or for any normal subgroup of G ?
Also, suppose there exists a homomorphism ϕ between G and G′. Let
H={x∈G;ϕ(x)∈H′}.
Then H is subgroup of G. We can also show that given that H′ is normal in G′, H is normal in G. Here, there exists an homomorphism between H and H′.
2. Is the function defining homomorphism between G and G′ same as H and H′ ?
From first isomorphism theorem, we can say, G/K≅G′ and H/K≅H′
3. Then Can I make this statement : Given a group G, and subgroup H of G, if there exists a homomorphism between G and G′ with Kernel K and H′ being a subgroup of G′, such that G/K≅G′ and H/K≅H′, then H is normal in G and H′ is normal in G′
Answers & Comments
Answer:
I understand the first isomorphic theorem, which states that a homomorphic image of a group is isomorphic to the quotient group formed by the group G and the kernel of group G.
1. Is this theorem only true for Kernel K, or for any normal subgroup of G ?
Also, suppose there exists a homomorphism ϕ between G and G′. Let
H={x∈G;ϕ(x)∈H′}.
Then H is subgroup of G. We can also show that given that H′ is normal in G′, H is normal in G. Here, there exists an homomorphism between H and H′.
2. Is the function defining homomorphism between G and G′ same as H and H′ ?
From first isomorphism theorem, we can say, G/K≅G′ and H/K≅H′
3. Then Can I make this statement : Given a group G, and subgroup H of G, if there exists a homomorphism between G and G′ with Kernel K and H′ being a subgroup of G′, such that G/K≅G′ and H/K≅H′, then H is normal in G and H′ is normal in G′
Explanation:
#CarryOnLearning