1. Mark each statement True or False: And explain your answer : (a) Let G be...

 

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1. Mark each statement True or False: And explain your answer : (a) Let G be a group and H be a subgroup of G. Then H is a subgroup of its normalizer N(H). (b) The dihedral group D, can be expressed as an internal direct product of two proper subgroups. (c) Every group of order 121 is an abelian group. (d) For a group G, if G/Z(G) is cyclic, then G/Z(G) must be the trivial group. (e) Let H= ((1 2 3)) and K = ((1 2)). Then H and K are normal subgroups of S3 and S3 is the internal direct product of H and K. (f) Let H = {[la ER-{0}}. Then H is a normal subgroup of GL(2, R). (g) Let G be a group with |G| = 24 and H be a subgroup of G with |H|= 12. Then H must be a normal subgroup of G. (h) For every prime p, the mapping f: Zp → Zp defined by f(x) = 2x is an isomorphism. 1. Mark each statement True or False: And explain your answer : (a) Let G be a group and H be a subgroup of G. Then H is a subgroup of its normalizer N(H). (b) The dihedral group D, can be expressed as an internal direct product of two proper subgroups. (c) Every group of order 121 is an abelian group. (d) For a group G, if G/Z(G) is cyclic, then G/Z(G) must be the trivial group. (e) Let H= ((1 2 3)) and K = ((1 2)). Then H and K are normal subgroups of S3 and S3 is the internal direct product of H and K. (f) Let H = {[la ER-{0}}. Then H is a normal subgroup of GL(2, R). (g) Let G be a group with |G| = 24 and H be a subgroup of G with |H|= 12. Then H must be a normal subgroup of G. (h) For every prime p, the mapping f: Zp → Zp defined by f(x) = 2x is an isomorphism.

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a True Since H is a group under the induced composition from G hence to prove that H is a subgroup o... View the full answer