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We show that for every $p\in (1,\infty )$ there exists a weight $w$ such that the Lorentz Gamma space ${\Gamma }_{p,w}$ is reflexive, its lower Boyd and Zippin indices are equal to zero and its upper Boyd and Zippin indices are equal to one. As a consequence, the Hardy-Littlewood maximal operator is unbounded on the constructed reflexive space ${\Gamma }_{p,w}$ and on its associate space ${\Gamma }_{p,w}^{\text{'}}$.