Nonattainment of the p-Jordan exponent in locally finite triangular groups

For every prime \\(p\\), we construct a countable locally finite triangular group \\(\\Gamma\\leq\\operatorname{GL}_2(\\overline{\\mathbb F}_p)\\) of derived length two whose admissible \\(p\\)-Jordan exponents are exactly \\[\\{s\\in\\mathbb R:\\exists J>0\\ \\forall H\\leq\\Gamma\\text{ finite},\\ j_p(H)\\leq J|H|_p^s\\}=(1,\\infty).\\] Here \\(j_p(H)\\) is the least index of a normal abelian \\(p^{\\prime}\\)-subgroup and \\(|H|_p\\) is the \\(p\\)-part of \\(|H|\\). Thus the exponent is one and is not attained, answering Kourovka Problem 21.121(a). The construction uses the exact multiplicative orders of divisors of \\(\\Phi_{k!}(p)\\) and the limit \\(\\varphi(k!)/k!\\to0\\).