Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 4 / 2 / ii

Table of contents
- Solution ii

Solution

Apply part (i) with

S = L

and use multiplicativity of the index of a subgroup along a subgroup chain:

[G : H \cap L] = [G : L] [L : H \cap L] \leq [G : L] [G : H] < \infty.

Thus the intersection of two finite-index subgroups again has finite index.

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