Let ø : G → G' be an isomorphism of a group (G, *) with a group(G', *'). Prove the following statements:a) If H is a subgroup of G, thenO[H] = {ó(h) |h€ H}is a subgroup of G'. In other words, an isomorphism carriessubgroups into subgroups.b) If G is cyclic, then G' is cyclic.

Let ø : G → G' be an isomorphism of a group (G, *) with a group (G', *'). Prove the following statements: a) If H is a subgroup of G, then O[H] = {«(h) |h € H} is a subgroup of G'. In other words, an isomorphism carries subgroups into subgroups. b) If G is cyclic, then G' is cyclic.

Let ø : G → G' be an isomorphism of a group (G, ) with a group (G', '). Prove the following statements: a) If H is a subgroup of G, then O[H] = {«(h) |h € H} is a subgroup of G'. In other words, an isomorphism carries subgroups into subgroups. b) If G is cyclic, then G' is cyclic.

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 ø : G → G' be an isomorphism of a group (G, *) with a group
(G', *'). Prove the following statements:
a) If H is a subgroup of G, then
O[H] = {ó(h) |h€ H}
is a subgroup of G'. In other words, an isomorphism carries
subgroups into subgroups.
b) If G is cyclic, then G' is cyclic.

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