On the TI-property and subnormality of self-centralizing non-metacyclic subgroups of a finite group
For self-centralizing non-metacyclic subgroups,we combine the TI-property and subnormality together to prove that if every self-centralizing non-metacyclic subgroup of a finite group G is a TI-subgroup or a subnormal subgroup,then every non-metacyclic subgroup of G is subnormal in G and such a group G is solvable.Moreover,we show that if every self-centralizing non-metacyclic subgroup of a finite group G is a TI-subgroup then every self-centralizing non-metacyclic subgroup of G is normal in G.
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