On infinite groups with abelian centralizers of involution

We obtain two characterizations of the projective linear group $\mathrm{PGL}_2(P)$ over a locally finite field $P$ of characteristic 2. The first is defined in terms of permutation groups; the second, in terms of the structure of the centralizers of involution. We use one of the characterizations to prove the existence of infinite groups recognizable by the set of their element orders.