Proof: Intersection of Subgroups is a Subgroup of H in G

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SUMMARY

The intersection of subgroups H and K of a group G, denoted as H ∩ K, is definitively a subgroup of H. This conclusion is based on the established definition of a subgroup, which states that a subset of a group is a subgroup if it is closed under the group operation and contains the identity element. Since H ∩ K is contained within H and is itself a group, it satisfies the criteria for being a subgroup of H.

PREREQUISITES
  • Understanding of group theory concepts, specifically subgroups.
  • Familiarity with set theory and its properties.
  • Knowledge of group operations and identity elements.
  • Basic proof techniques in abstract algebra.
NEXT STEPS
  • Study the definition and properties of subgroups in group theory.
  • Explore examples of subgroup intersections in finite groups.
  • Learn about the Lattice Theorem in group theory.
  • Investigate the implications of subgroup properties in larger algebraic structures.
USEFUL FOR

Students of abstract algebra, mathematicians focusing on group theory, and anyone interested in the structural properties of groups and subgroups.

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Homework Statement


If H, K are subgroups of G, show that H intersect K is a subgroup of H

Homework Equations


I know that H intersect K is a subgroup of G; I proved this already but I'm wondering how H intersect K is a subgroup of H

The Attempt at a Solution


I'm quite sure this is true but my idea is based on set theoretic properties and one can't use such properties to apply on subgroups
 
Physics news on Phys.org
If you've proved H intersect K is a group, then there is nothing more to prove. H intersect K is contained H and it's a group. Therefore it's a subgroup of H. That's the definition of 'subgroup'.
 

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