Hey everyone, I've got a question in elementary group theory.
Suppose we have a group G, and we want to completely partition it into multiple subgroups, such that the only element each subgroup shares with any other is the identity element. Is this ever possible?
I think that such a partitioning is impossible for any cyclic group. If g is a generator for a cyclic group G, and H is a subgroup of G with a smaller order than G, then g [itex] \notin [/itex] H, because closure of H would require all of g's powers (and hence all of G) to be in H. Therefore we cannot include the generators of G in any of the partitions, so partitioning is impossible.
What about non-cyclic groups, like the Klein Four Group?
Thanks!
Suppose we have a group G, and we want to completely partition it into multiple subgroups, such that the only element each subgroup shares with any other is the identity element. Is this ever possible?
I think that such a partitioning is impossible for any cyclic group. If g is a generator for a cyclic group G, and H is a subgroup of G with a smaller order than G, then g [itex] \notin [/itex] H, because closure of H would require all of g's powers (and hence all of G) to be in H. Therefore we cannot include the generators of G in any of the partitions, so partitioning is impossible.
What about non-cyclic groups, like the Klein Four Group?
Thanks!
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