Lemmas of Fixing Subgroups

Here we outline a few lemmas of Fixing Subgroups:

Lemma

For any collection ${H_{i}}$ of Subgroups of a Group the intersection:

⋂_{i} H_{i}

is a Subgroup of $G$ . (it's the largest subgroup inside all $H_{i}$ )

Lemma

For subsets $S_{1} \subseteq S_{2} \subseteq X$ we have:

Fix (X) \leq Fix (S_{2}) \leq Fix (S_{1}) \leq Fix (\emptyset) = G