Suppose H and K are both subgroup of a finite group G, and H⊂K. Prove [G:H]=[G:K][K:H]
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Originally Posted by apple2009 Suppose H and K are both subgroup of a finite group G, and H⊂K. Prove [G:H]=[G:K][K:H] In a sense here, we see that . It follows by Lagrange's Theorem that both and divide . Therefore, . Substituting the third equation into the first, we get . Now substituting this result into the second equation, we have . Does this make sense?
a less trivial version of this problem is to let G be any group (not necessarily finite) and assume that [G : H] is finite. note that then both [G : K] and [K : H] will also be finite.
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