Index satisfies intersection inequality

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Statement

Suppose G is a group and H, K are subgroups of finite index in G. Then, we have:

[G:H \cap K] \le [G:H][G:K].

(An analogous statement holds for subgroups of infinite index, provided we interpret the indices as infinite cardinals).

Related facts

Facts used

  1. Index satisfies transfer inequality: This states that if H, K \le G, then [K: H \cap K] \le [G:H].
  2. Index is multiplicative: This states that L \le K \le G, then [G:L] = [G:K][K:L].

Proof

Given: A group G with subgroups H and K.

To prove: [G:H \cap K] \le [G:H][G:K].

Proof: By fact (1), we have:

[K:H \cap K] \le [G:H].

Setting L = H \cap K in fact (2) yields:

[G:H \cap K] = [G:K][K:H \cap K].

Combining these yields:

[G:H \cap K] \le [G:H][G:K]

as desired.