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Contranormal subgroup information


In mathematics, in the field of group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group.[1] Clearly, a contranormal subgroup can be normal only if it is the whole group.

Some facts:

  • Every subgroup of a finite group is a contranormal subgroup of a subnormal subgroup. In general, every subgroup of a group is a contranormal subgroup of a descendant subgroup.
  • Every abnormal subgroup is contranormal.
  1. ^ Rose 1968, p. 97

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Contranormal subgroup

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group theory, a contranormal subgroup is a subgroup whose normal closure in the group is the whole group. Clearly, a contranormal subgroup can be normal...

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Normal subgroup

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subgroup Contranormal subgroup Abnormal subgroup Self-normalizing subgroup Characteristic subgroup Fully characteristic subgroup Subnormal subgroup Ascendant...

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Abnormal subgroup

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a contranormal subgroup. The only normal subgroup that is also abnormal is the whole group. Every abnormal subgroup is a weakly abnormal subgroup, and...

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Glossary of group theory

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conjugate subgroups Two subgroups H1 and H2 of a group G are conjugate subgroups if there is a g ∈ G such that gH1g−1 = H2. contranormal subgroup A subgroup of...

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