Answer each question as a form of discussion: 1) Let be a characteristic subgroup of . Then, is normal in . Characteristic subgroup A subgroup of a group is termed a characteristic subgroup, if for...

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Answered Same DayDec 29, 2021

Answer To: Answer each question as a form of discussion: 1) Let be a characteristic subgroup of . Then, is...

David answered on Dec 29 2021
119 Votes
Characteristic subgroup: If (G ) is a group, then H is a characteristic subgroup of
G if every Aut
omorphism of G maps H to itself i.e., if f Aut(G) and h H then f(h)
H .
Some properties of characteristic subgroups are
1) If G has only one subgroup of a given cardinality then that subgroup is
characteristic.
2) If H char G then H is a normal subgroup of G.
3) If K char H and H G then K G.
If H char G and K char H and then K char G. In other words, any subgroup
obtained by taking the image of this subgroup under an automorphism of the whole
group, must be equal to it. A subgroup is characteristic if and only if it is the union of
equivalence classes of elements under the action of the automorphism group. Not
every normal subgroup is characteristic. If H is a fully characteristic subgroup of K,
and K is a fully characteristic subgroup of G, then H is a fully characteristic subgroup
of G. it is true that every fully characteristic...
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