What groups have exactly 4 subgroups?

In summary, the conversation discusses the classification of groups with a certain number of subgroups. The participants mention that any nontrivial group has at least two subgroups, and that a subgroup of a subgroup is a subgroup of the original group. They also mention Cauchy's theorem and Sylow's theorem, which states that there is a subgroup of order p^k whenever p is prime and p^k divides the order of the group. It is suggested that groups of order pq and cyclic groups of order p^3 may satisfy the conditions, but it is uncertain if a group of order p^3 would work.
  • #1
Mystic998
206
0
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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  • #2
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.
 
  • #3
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...
 
  • #4
Yeah, I think groups of order pq work. Also, I think a cyclic group of order p^3 works...
 

FAQ: What groups have exactly 4 subgroups?

What is the purpose of classifying groups?

Classifying groups is a way to organize and categorize different types of objects or entities based on their similarities and differences. This allows us to better understand and study the characteristics and relationships within a group.

How are groups classified?

Groups can be classified in multiple ways, such as by their physical characteristics, behaviors, functions, or relationships. The most common method of classification is through a hierarchical system, where groups are organized into larger categories and subcategories.

What are the main criteria for classifying groups?

The main criteria for classifying groups include shared characteristics, common ancestry, and evolutionary relationships. Other factors such as habitat, behavior, and function may also be considered depending on the specific classification system being used.

Can groups be reclassified?

Yes, groups can be reclassified if new information or evidence is discovered that changes our understanding of their relationships or characteristics. This is a common occurrence in the scientific community as new research and discoveries are constantly being made.

What are the limitations of classification?

Classification is a useful tool for organizing and understanding groups, but it is not a perfect system. It can be limited by the criteria and methods used, as well as the constantly evolving nature of science. Additionally, some groups may not fit neatly into one category or may have characteristics that overlap with other groups, making classification more challenging.

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