What groups have exactly 4 subgroups?

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Discussion Overview

The discussion revolves around the classification of groups that possess exactly four subgroups. Participants explore theoretical aspects of group theory, particularly focusing on the implications of group order and subgroup structure.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant expresses uncertainty about how to classify groups with exactly four subgroups, indicating a lack of clarity on the topic.
  • Another participant notes that any nontrivial group must have at least two subgroups: the group itself and the trivial subgroup containing only the identity.
  • A participant references Cauchy's theorem and Sylow's theorem, suggesting that groups of order pq (where p and q are primes) may meet the criteria for having exactly four subgroups.
  • There is a discussion about the possibility of groups of order p^3 also working, although one participant expresses skepticism about this scenario.

Areas of Agreement / Disagreement

Participants appear to agree that groups of order pq may satisfy the condition of having exactly four subgroups, but there is uncertainty regarding the potential for groups of order p^3. The discussion remains unresolved regarding the complete classification of such groups.

Contextual Notes

Participants have not fully explored the implications of subgroup structure or the specific conditions under which groups of higher orders may or may not fit the criteria. There are also unresolved assumptions about the nature of the groups being discussed.

Who May Find This Useful

This discussion may be of interest to those studying group theory, particularly in the context of subgroup classification and theorems related to group orders.

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

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