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 any automorphism of , we have (H) = H. We can write characteristicity as the invariance property with respect to automorphisms:
Characteristic = Automorphism ? Function
This is interpreted as: any automorphism from the whole group to itself, restricts to a function from the subgroup to itself. In other words, the subgroup is invariant under automorphisms. We can also write characteristicity as:
Automorphic subgroups ? Equal
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.
Normal subgroup
A subgroup of a group is termed normal, if for any , the inner automorphism defined by conjugation by , namely the map x ?? gxg-1, gives an isomorphism on . In other words, for any :
(H) = H
or more explicitly:
Implicit in this definition is the fact that is an automorphism. We can write normality as the invariance property with respect to inner automorphisms:
Normal = Inner automorphism ? Function
In other words, any inner automorphism on the whole group restricts to a function from the subgroup to itself. We can also write:
Normal = Conjugate subgroups ? Equal
In other words, any subgroup conjugate to the given one, must be equal to it. A subgroup is normal if and only if it is a union of conjugacy classes of elements.
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2) A a composition series is a group or a module broken into simple groups/modules. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
A composition series may not even exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan-Hölder theorem asserts that whenever composition series exist, the isomorphism is into simple pieces (although, perhaps, not their location in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of finite groups and Artinian modules.
From http://en.wikipedia.org/wiki/Composition_series
An example of a composition series for the quaternion qroup Q.is
Quaterion qroup of order 8:
Q: = { +/-1, +/-i, +/-j, +/- k} c Hx
Classes is 1, -1, i x 2, j x 2, k 2.
[Q, Q] = Z(q) = <-1>, Q/[Q,Q] = V.
http://livetoad.org/Courses/Documents/4f10/Notes/character_tables_ XXXXXXXXXXpdf