Cyclic normal groups

  • Thread starter math8
  • Start date
  • #1
160
0
Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G.

I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have
K char in H and H normal in G.
Hence K normal in G.

I am not sure about the part where subgroups of cyclic groups are characteristic. If yes, How would you prove this?
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,263
619
Think concretely. A cyclic group is isomorphic to either the integers Z, or the integers mod n, Z_n. Can you prove any subgroup of those is characteristic?
 
  • #3
160
0
I am thinking maybe that since cyclic groups of a certain order are unique up to isomorphism and that if a subgroup K of a certain order is unique in a group H, then K char in G.

Now since K is cyclic in H, then K char in H.
 
  • #4
Dick
Science Advisor
Homework Helper
26,263
619
I am thinking maybe that since cyclic groups of a certain order are unique up to isomorphism and that if a subgroup K of a certain order is unique in a group H, then K char in G.

Now since K is cyclic in H, then K char in H.

Something like that. If you can prove there is exactly one subgroup of a given order in Z_n then you've got it. In the infinite case of Z, it's not going to be useful to consider order though.
 

Related Threads on Cyclic normal groups

Replies
11
Views
7K
  • Last Post
Replies
1
Views
966
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
9
Views
6K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
6
Views
2K
Top