Classification of groups

Mystic998

I was wondering about the classification of groups with a certain number of subgroups. I (sort of mostly I think maybe) get the ideas behind classification of groups of a certain (hopefully small) order, but I came across a question about classifying all groups with exactly 4 subgroups, and I have no clue whatsoever how to even begin.

morphism

Well, any (nontrivial) group has at least two subgroups: itself and the subgroup consisting of only the identity. Moreover, a subgroup of a subgroup is a subgroup of the original group. Another thing you might find helpful is Cauchy's theorem.

mathwonk

sylow's theorem says there is a subgroup of order p^k whenever p is prime and p^k divides the order of the group. so groups of order pq seem to satisfy the condition. where p and q are prime. but there canniot be more than two factors of the order of the group, and there cannot be a factor occurring to a power higher than 2? or could a group of order p^3 possibly work? seems unlikely........

Mystic998

Yeah, I think groups of order pq work. Also, I think a cyclic group of order p^3 works...

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Mystic998	Classification of groups Tweet I was wondering about the classification of groups with a certain number of subgroups. I (sort of mostly I think maybe) get the ideas behind classification of groups of a certain (hopefully small) order, but I came across a question about classifying all groups with exactly 4 subgroups, and I have no clue whatsoever how to even begin.

morphism Recognitions: Homework Help Science Advisor	Well, any (nontrivial) group has at least two subgroups: itself and the subgroup consisting of only the identity. Moreover, a subgroup of a subgroup is a subgroup of the original group. Another thing you might find helpful is Cauchy's theorem.

mathwonk Recognitions: Homework Help Science Advisor	sylow's theorem says there is a subgroup of order p^k whenever p is prime and p^k divides the order of the group. so groups of order pq seem to satisfy the condition. where p and q are prime. but there canniot be more than two factors of the order of the group, and there cannot be a factor occurring to a power higher than 2? or could a group of order p^3 possibly work? seems unlikely........

Mystic998	Classification of groups Yeah, I think groups of order pq work. Also, I think a cyclic group of order p^3 works...