Let H, K be groups and α : K → Aut(H) be a homomorphism of groups....

Let H, K be groups and α : K → Aut(H) be a homomorphism of groups. Show that H oα K is the internal semidirect product of subgroups which are isomorphic to H and K, respectively

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Colby Messinger answered 2 years ago

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K...

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K then G ≅ K.

Let f be a group homomorphism from a group G to a group H If the...

Let f be a group homomorphism from a group G to a group H If the order of g equals the order of f(g) for every g in G must f be one to one.

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is...

Let φ : G1 → G2 be a group homomorphism. (abstract algebra) (a) Suppose H is a subgroup of G1. Define φ(H) = {φ(h) | h ∈ H}. Prove that φ(H) is a subgroup of G2. (b) Let ker(φ) = {g ∈ G1 | φ(g) = e2}. Prove that ker(φ) is a subgroup of G1. (c) Prove that φ is a group isomorphism if and only if ker(φ) = {e1} and φ(G1) = G2.

Let f : G → G′ be a surjective homomorphism between two groups, G and G′,...

Let f : G → G′ be a surjective homomorphism between two groups, G and G′, and let N be a normal subgroup of G. Prove that f (N) is a normal subgroup of G′.

Direct product of groups: Let (G, ∗G) and (H, ∗H) be groups, with identity elements eG...

Direct product of groups: Let (G, ∗G) and (H, ∗H) be groups, with identity elements eG and eH, respectively. Let g be any element of G, and h any element of H. (a) Show that the set G × H has a natural group structure under the operation (∗G, ∗H). What is the identity element of G × H with this structure? What is the inverse of the element (g, h) ∈ G × H? (b) Show that the map...

Let G, H be groups and define the relation ∼= where G ∼= H if there...

Let G, H be groups and define the relation ∼= where G ∼= H if there is an isomorphism ϕ : G → H. (i) Show that the relation ∼= is an equivalence relation on the set of all groups. (ii) Give an example of two different groups that are related.

Let M/F and K/F be Galois extensions with Galois groups G = Gal(M/F) and H =...

Let M/F and K/F be Galois extensions with Galois groups G = Gal(M/F) and H = Gal(K/F). Since M/F is Galois, and K/F is a field extension, we have the composite extension field K M. Show that σ → (σ|M , σ|K) is a homomorphism from Gal(K M/F) to G × H, and that it is one-to-one. [As in the notes, σ|X means the restriction of the map σ to the subset X of its domain.]

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Let H, K be groups and α : K → Aut(H) be a homomorphism of groups....

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