Math Help for Finite Cyclic Group & Subgroups

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Discussion Overview

The discussion revolves around problems related to group theory, specifically focusing on finite cyclic groups, subgroups, and properties of group elements. The scope includes theoretical questions and mathematical reasoning regarding group structures and subgroup relationships.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a problem involving a finite cyclic group of order p^n and questions whether either of two subgroups H or K must contain the other, expressing uncertainty about their approach.
  • Another participant suggests that since G is cyclic, the elements in H and K can be analyzed in terms of their orders, implying a relationship between the subgroups.
  • In response to the second problem, a participant argues against using cosets and instead focuses on the intersection of subgroups of order 5, suggesting that if there are 7 distinct subgroups, they must contain a limited number of elements.
  • One participant challenges the clarity of the third problem, indicating confusion about its formulation.
  • A suggestion is made to consider small non-abelian groups when discussing the fourth problem about subgroups of index 3, hinting at the symmetric group S3 as a potential example.
  • Another participant reflects on problem 3, proposing the use of the alternating group A6 and considering its subgroups and cosets.
  • A later post corrects the statement of problem 2, clarifying that it involves the intersection of all subgroups of G being non-trivial, and poses a question about elements of infinite order.
  • A participant introduces a hypothetical element of infinite order and discusses the cyclic groups generated by powers of this element, questioning the nature of their intersection.

Areas of Agreement / Disagreement

Participants express various viewpoints on the problems, with some indicating confusion or disagreement about the formulations and approaches. No consensus is reached on the solutions or interpretations of the problems presented.

Contextual Notes

There are limitations in the clarity of problem statements, particularly in the third problem, which some participants find confusing. Additionally, the discussions involve assumptions about subgroup properties that may not be universally applicable.

Who May Find This Useful

Undergraduate students studying group theory, educators looking for examples of group properties, and participants interested in mathematical reasoning related to abstract algebra.

cauchys_pet
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hey! great to find such an informative website...
i'm an undergrad math student and have lots of problems with group theory.i hope you'll all help me enjoy group theory...
my teacher put forward these question last week and I've been breaking my head over them without much success :
1. let G be a finite cyclic group of order p^n, p being prime and n >=0. if H and K are subgroups of G then show that either H contains K or K contains H.
i started out supposing the contrary but i wonder if I'm on the right track. i don't think it helps. :confused:

2.if G is a group of order 30 show that G has atmost 7 distinct subgroups of order 5.
can i say this : let H be a subgroup of order 5 then the number of distinct left cosets of H in G is 6. so are we done?!

3.let G be a group such that intesection of all subgroups of G different from {e}. then prove that every element of G has finite order.

4. give an example to show that a subgroup of index 3 may not be a normal subgroup of G. :frown:
thanks again for the help.
 
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1. G is cyclic what does that mean? so what can you say about the elements in H and K in terms of this?


2. Is nothing to do with cosets. Suppose H and K are subgroups of order 5, then HnK is a subgroup whose order divides 5, so it follows HnK=H=K or HnK={e} So the subgroups are either equal or contain only one element in common. So if there 7 (or fewer) distinct subgroups of order 5 these contain 7*4+1=29 disticnt elements: they all contain e, and 4 other elements each that appear in exactly one subgroup. If there were more than 7 then what would happen?

3 makes no sense.

4. Hmm, can you think of any small subgroups that have a subgroup of index 3 that aren't abelian? Try the smallest such (it has order 6...)
 
have been thinking about prob 3...i guess you take A6 (the set of even permutations of 6 elements), possibly find a subgroup H of order 4 and then look at the 3 distinct cosets of H.
 
sorry, i goofed up problem number 2
it says, let G be a group such that intersection of all subgroups of G different from {e} is not {e}. prove that every element is of finite order.
 
suppose there is an element of infinite order, g say.

For all r in N let C_r be the cyclic group generated by g^r,

What is the intersection of all these?
 

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