Solved: Prove |HK|=|H||K| When H, K are Subgroups of G and H\capK = <e>

Thread starter chycachrrycol
Start date Mar 27, 2011

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SUMMARY

The discussion centers on proving that if H and K are subgroups of a group G, and their intersection H ∩ K is trivial (i.e., contains only the identity element ), then the order of the product of the two subgroups satisfies the equation |HK| = |H||K|. The proof involves demonstrating that each element of HK can be uniquely expressed as a product of elements from H and K, leveraging the properties of group operations and subgroup definitions.

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Understanding of group theory concepts, specifically subgroups and their properties.
Familiarity with the notation and operations involving groups, such as the product of subgroups.
Knowledge of the identity element in group theory.
Experience with basic proofs in abstract algebra.

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Mar 27, 2011

chycachrrycol

So the problem is if H and K are subgroups of G with HK = {hk \in G| h \in H, k \in K}. If we know that H\capK = <e>, show |HK|= |H||K|

My work so far:
h, j \in H
k,l \in K
i know that if hk = jl then j^{-1}h = lk^{-1}
But I'm not sure what to do from here.