Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = ( (A)). Here (A) denotes the subgroup generated by A. (b) Let E = {a € G : (a) = v (a)} . Prove E G.
Let G and H be two groups, , & € Hom (G, H), and A C G be a nonempty subset. (a) Prove ((A)) = ( (A)). Here (A) denotes the subgroup generated by A. (b) Let E = {a € G : (a) = v (a)} . Prove E G.
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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![Let \( G \) and \( H \) be two groups, \( \phi, \psi \in \text{Hom}(G, H) \), and \( A \subset G \) be a nonempty subset.
(a) Prove
\[ \phi(\langle A \rangle) = \langle \phi(A) \rangle. \]
Here, \( \langle A \rangle \) denotes the subgroup generated by \( A \).
(b) Let
\[ E = \{ a \in G : \phi(a) = \psi(a) \}. \]
Prove \( E \leq G \).
Prove any subgroup of a cyclic group is itself a cyclic group.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97e612ef-1556-436b-b62c-352b280e9e69%2Fb395ed9c-5d6b-4d92-a79d-ef6c2c806758%2Fs6rnhxl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let \( G \) and \( H \) be two groups, \( \phi, \psi \in \text{Hom}(G, H) \), and \( A \subset G \) be a nonempty subset.
(a) Prove
\[ \phi(\langle A \rangle) = \langle \phi(A) \rangle. \]
Here, \( \langle A \rangle \) denotes the subgroup generated by \( A \).
(b) Let
\[ E = \{ a \in G : \phi(a) = \psi(a) \}. \]
Prove \( E \leq G \).
Prove any subgroup of a cyclic group is itself a cyclic group.
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Step 1
We have proved (a) in two steps, showing both the subset relation one by one.
For (b) we have used the theorem that to prove H is a subgroup of G we need to prove ab^{-1} belong to H for all a,b in H.
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