If there exist a non cyclic group $G$ with all sylow $p$subgroups cyclic,and the normal $p\_1$-complement $M$ for $G$ is cyclic,here $p\_1$ is the smallest factor of $|G|$?And when does it always exist?



a group with all sylow p subgroups cyclic 
There is a complete classification of groups with all Sylow-subgroups being cyclic. In fact one can weaken this: we say that a group $G$ is **almost Sylow-cyclic** if every Sylow subgroup of $G$ has a cyclic subgroup of index at most $2$. Almost Sylow-cyclic groups are fully classified in two papers:



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> M. Suzuki, On finite groups with cyclic Sylow subgroups for all odd primes, Amer. J. Math. 77 (1955) 657–691.
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> W.J. Wong, On finite groups with semi-dihedral Sylow 2-subgroups, J. Algebra 4 (1966) 52–63.
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You may also be interested in an old paper by Holder from 1895 who proved
that every group with all Sylow subgroups cyclic is solvable. (This is not true under the weaker supposition that a group is almost Sylow-cyclic, as the group $PSL\_2(7)$ demonstrates.)


