Series (group theory)

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In group theory, a series is a chain (mathematics) of subgroups of a group ordered by subset inclusion. The structure of the group is closely related to the existence of series with particular properties.

A series is a linearly ordered chain of subgroups of a given group G beginning with the group G itself:

G = A_0 \supseteq A_1 \supseteq \cdots \supseteq A_n . \,

If the final group in the series is H we speak of a series from G to H.

The series is subinvariant or subnormal if each subgroup is a normal subgroup of its predecessor, A_i \triangleleft A_{i-1}. A subinvariant series in which each subgroup is a maximal normal subgroup of its predecessor is a composition series.

The series is invariant or normal if each subgroup is a normal subgroup of the whole group. A subinvariant series in which each subgroup is a normal subgroup of G maximal subject to being a proper subgroup of its predecessor is a principal series or chief series.

References

  • Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 123-124. 
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