Is the word you're looking for "cavalry"? This is the general term for mounted troops. (In modern armies, it sometimes refers to mechanized divisions... i.e. tanks... rather than soldiers on actual horses.)
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During the US Civil War there were two names given to soldiers who rode on horses. The traditional cavalry was augmented by what was called "mounted infantry". This was due to the fact that infantry soldiers who could ride a horse were engaged in battles. This mounted infantry would dismount and act as infantry soldiers. The horses were a means to get the soldiers to a battle quickly.