1. The problem statement, all variables and given/known data
Let [itex] G [/itex] be a finite group where [itex] H [/itex] and [itex] K [/itex] are subgroups of [itex] G [/itex]. Prove that [itex] |HK|=\frac{|H||K|}{|H \cap K|} [/itex].
2. Relevant equations
set [itex] HK=\{x\in G| x=st, s\in H and t\in K\}[/itex]
3. The attempt at a solution
I am a bit lost with this problem. What I did was break this proof into two cases. Since H and K are subgroups then [itex]|H \cap K|[/itex] is a subgroup. So case 1: [itex]|H \cap K|=1[/itex] Thus the only common element between H and K is the identity element call it e. It follows that e repeats only once in set [itex] HK [/itex]. Thus [itex] |HK|=\frac{|H||K|}{|H \cap K|} [/itex]. For case 2 Im lost here where [itex]|H \cap K|>1[/itex]. I'm not sure if case 1 is correct it seems correct but for case 2 I need some help. I know case 2 is similar to case 1. So I was thinking let [itex] r=|H \cap K| [/itex]. Then I know that [itex] r[/itex] repeats r times in [itex] HK [/itex] since those are the only elements in common between H and K. Plus [itex] H [/itex]and[itex] K [/itex]are nonempty since their subgroups. So that means that there are r of these elements in [itex] H[/itex] and [itex] K[/itex]. Thus [itex] r| |H||K|[/itex]. These are my ideas but I don't know how to put them together. Thanks.
Let [itex] G [/itex] be a finite group where [itex] H [/itex] and [itex] K [/itex] are subgroups of [itex] G [/itex]. Prove that [itex] |HK|=\frac{|H||K|}{|H \cap K|} [/itex].
2. Relevant equations
set [itex] HK=\{x\in G| x=st, s\in H and t\in K\}[/itex]
3. The attempt at a solution
I am a bit lost with this problem. What I did was break this proof into two cases. Since H and K are subgroups then [itex]|H \cap K|[/itex] is a subgroup. So case 1: [itex]|H \cap K|=1[/itex] Thus the only common element between H and K is the identity element call it e. It follows that e repeats only once in set [itex] HK [/itex]. Thus [itex] |HK|=\frac{|H||K|}{|H \cap K|} [/itex]. For case 2 Im lost here where [itex]|H \cap K|>1[/itex]. I'm not sure if case 1 is correct it seems correct but for case 2 I need some help. I know case 2 is similar to case 1. So I was thinking let [itex] r=|H \cap K| [/itex]. Then I know that [itex] r[/itex] repeats r times in [itex] HK [/itex] since those are the only elements in common between H and K. Plus [itex] H [/itex]and[itex] K [/itex]are nonempty since their subgroups. So that means that there are r of these elements in [itex] H[/itex] and [itex] K[/itex]. Thus [itex] r| |H||K|[/itex]. These are my ideas but I don't know how to put them together. Thanks.
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