3 Let N be a normal subgroup of G. Suppose Nn [G,G] = 1. Show that N < Z(G). group. -] Show that every quotient group of a nilpotent group is also a nilpotent
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![3 Let N be a normal subgroup of G. Suppose Nn [G,G] = 1. Show that
N < Z(G).
group.
-] Show that every quotient group of a nilpotent group is also a nilpotent](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fefed8592-128e-41f1-b41a-16ede2a8f456%2F042caa14-c6f9-42cd-a7a1-96ba98a7d01e%2Fqh9e8a9_processed.jpeg&w=3840&q=75)
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- Find all subgroups of the quaternion group.Find the right regular representation of G as defined Exercise 11 for each of the following groups. a. G={ 1,i,1,i } from Example 1. b. The octic group D4={ e,,2,3,,,, }.18. If is a subgroup of the group such that for all left cosets and of in, prove that is normal in.
- Show that every subgroup of an abelian group is normal.22. If and are both normal subgroups of , prove that is a normal subgroup of .Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=40