If G is a finite group, we define its prime graph Γ(G) as follows. The vertices of Γ(G) are the primes dividing the
order of G and two distinct vertices p, q are joined by an edge, denoted by p~q, if and only if there is an element in G of order pq (see [7, 15]). Assume |G|=p1α1p2α2…pkαk with primes p1<p2<…<pk and natural numbers αi. For p∈π(G)={p1, p2, … , pk, define deg(p):=|{q∈π(G)|q~p}|, which is called the degree of p. We also define D(G):=(deg(p1), deg(p2), …, deg(pk)), which is called the degree pattern of the group G. We say a group G is t-fold OD-characterizable if there exist exactly t non-isomorphic finite groups M such that |M|=|G| and D(M)=D(G) (see [11]). In particular, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In the present paper, we prove that the projective special linear group L15(2) is OD-characterizable by a newly introduced lemma to deal with its connected prime graph. As a consequence of this result, we obtain that L15(2) is OG-characterizable.
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