Abstract Let n be a positive integer. We say that a group G is an ( n + 1 2 ) {(n+\frac{1}{2})} -Engel group if it satisfies the law [ x , y n , x ] = 1 {[x,{}_{n}y,x]=1} . The variety of ( n + 1 2 ) {(n+\frac{1}{2})} -Engel groups lies between the varieties of n-Engel groups and ( n + 1 ) {(n+1)} -Engel groups. In this paper, we study these groups, and in particular, we prove that all ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel { 2 , 3 } {\{2,3\}} -groups are locally nilpotent. We also show that if G is a ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel p-group, where p ≥ 5 {p\geq 5} is a prime, then G p {G^{p}} is locally nilpotent.

中文翻译：

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在 (𝑛 + ½)-恩格尔群

摘要 令 n 为正整数。我们称群 G 是 ( n + 1 2 ) {(n+\frac{1}{2})} -Engel 群，如果它满足定律 [ x , yn , x ] = 1 {[x,{} _{n}y,x]=1} 。( n + 1 2 ) {(n+\frac{1}{2})} -Engel 群的种类介于 n-Engel 群和 ( n + 1 ) {(n+1)} -Engel 群之间. 在本文中，我们研究了这些群，特别是我们证明了所有 ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel { 2 , 3 } {\{2,3 \}} -groups 是局部幂零的。我们还表明，如果 G 是 a ( 4 + 1 2 ) {(4+\frac{1}{2})} -Engel p-group，其中 p ≥ 5 {p\geq 5} 是素数，则 G p {G^{p}} 是局部幂零的。