Abstract Algebra- homomorphisms and Isomorphisms, proving not cyclic

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Homework Help Overview

The discussion revolves around proving properties of subgroups in the context of abstract algebra, specifically focusing on homomorphisms and isomorphisms. The original poster presents a problem involving distinct subgroups of a group G, their intersection, and the nature of the quotient group formed.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to prove that the intersection of two distinct subgroups is a normal subgroup and to determine the index of this intersection. They express uncertainty about how to show the index is 4 and how to approach the second part regarding the quotient group not being cyclic.
  • Some participants question the nature of the product of the subgroups and suggest considering the properties of the quotient group, particularly its subgroups of order 2.
  • Others note the importance of understanding cosets in relation to the quotient group and suggest examining specific elements and their cosets.

Discussion Status

The discussion is active, with participants exploring different aspects of the problem. Some have provided hints and guidance regarding the use of subgroup properties and the structure of quotient groups. There is a focus on clarifying concepts rather than reaching a consensus or conclusion.

Contextual Notes

Participants are navigating through the complexities of subgroup indices and properties of quotient groups, with some expressing confusion about the definitions and implications of these concepts. The original poster references a textbook for guidance, indicating reliance on external resources for problem-solving.

eruth
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1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic.


2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K is a subgroup of G and N is a normal subgroup of G, then K/(K intersect N) is isomorphic to KN/N

3. The Attempt at a Solution - I know that any subgroup with order 2 is normal and that the intersection of 2 normal subgroups is normal. I just wasn't sure how to show it was index 4. Also, I wasn't what to do for the second part.
 
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Hint: what do you know about the product HK?
 
Alright, I think I got the first part of the proof. Now to show that the quotient group isn't cyclic, the back of my book says to observe that it has 2 subgroups of order 2... I'm not seeing what these 2 subgroups are, maybe just because quotients groups confuse me.
 
eruth said:
Alright, I think I got the first part of the proof. Now to show that the quotient group isn't cyclic, the back of my book says to observe that it has 2 subgroups of order 2... I'm not seeing what these 2 subgroups are, maybe just because quotients groups confuse me.

Define L=HnK. You should know that the quotient group is defined by the products of cosets. Take h to be an element of H that's not in L. What coset could (hL)(hL) be?
 

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