Geodesic
Groups with Metacyclic Sylow 2-Subgroups
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1234-1237

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A group S is said to be metacyclic if it contains a normal cyclic subgroup N such that S/N is cyclic. In this note the following theorem is proved.THEOREM. Let G be a group, S a metacyclic Sylow 2-subgroup of G. If S has a cyclic normal subgroup N such that S/N is cyclic of order greater than 2, then G is soluble. Remark. We show that such a group G contains a 2-nilpotent normal subgroup of index a divisor of 6. The solubility of these groups requires the solubility of groups of odd order unavoidably. Notation. All groups considered will be finite. Let G be a group, S a subset of G, Aand B subgroups of G, N a normal subgroup of G.〈S〉: the subgroup of G generated by S. NG (S): the normalizer of S in G. CG (S): the centralizer of S in G. 
Camina, A. R.; Gagen, T. M. Groups with Metacyclic Sylow 2-Subgroups. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1234-1237. doi: 10.4153/CJM-1969-135-3
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