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If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element a EG, define p.: X-X as follows: P.(xH) = (ax)H 1 Prove that each p, is a permutation of X. 2 Rrove that h: G-Sy defined by h(a)=p, is a homomorphism # 3/Prove that the set (a E H: xax-¹E H for every x E G}), that is, the set of all the elements of H whose conjugates are all in H, is the kernel of h. Prove that if H contains no normal subgroup of G except (e), then G is isomorphic to a subgroup of S

If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element a EG, define p.: X-X as follows: P.(xH) = (ax)H 1 Prove that each p, is a permutation of X. 2 Rrove that h: G-Sy defined by h(a)=p, is a homomorphism # 3/Prove that the set (a E H: xax-¹E H for every x E G}), that is, the set of all the elements of H whose conjugates are all in H, is the kernel of h. Prove that if H contains no normal subgroup of G except (e), then G is isomorphic to a subgroup of S

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element a E G, define p.: X→ X as follows: P.(xH) = (ax)H 1 Prove that each p, is a permutation of X. 2 Prove that h: G-Sy defined by h(a)=p, is a homomorphism. # 3/Prove that the set {a E H: xax-¹E H for every x E G), that is, the set of all the elements of H whose conjugates are all in H, is the kernel of h. Prove that if H contains no normal subgroup of G except {e}, then G is isomorphic to a subgroup of Sy
Transcribed Image Text:If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element a E G, define p.: X→ X as follows: P.(xH) = (ax)H 1 Prove that each p, is a permutation of X. 2 Prove that h: G-Sy defined by h(a)=p, is a homomorphism. # 3/Prove that the set {a E H: xax-¹E H for every x E G), that is, the set of all the elements of H whose conjugates are all in H, is the kernel of h. Prove that if H contains no normal subgroup of G except {e}, then G is isomorphic to a subgroup of Sy
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