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.

Answered: Let G and H be two groups, , & € Hom…

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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Question

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.

Expert Solution

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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