Schreier refinement theorem
Encyclopedia
In mathematics
, the Schreier refinement theorem of group theory
states that any two normal series
of subgroup
s of a given group have equivalent refinements.
The theorem is named after the Austria
n mathematician
Otto Schreier
who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Schreier refinement theorem of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
states that any two normal series
Normal series
In mathematics, a subgroup series is a chain of subgroups:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G.Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups...
of subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s of a given group have equivalent refinements.
The theorem is named after the Austria
Austria
Austria , officially the Republic of Austria , is a landlocked country of roughly 8.4 million people in Central Europe. It is bordered by the Czech Republic and Germany to the north, Slovakia and Hungary to the east, Slovenia and Italy to the south, and Switzerland and Liechtenstein to the...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Otto Schreier
Otto Schreier
Otto Schreier was an Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups. He studied mathematics at the University of Vienna and obtained his doctorate in 1923, under the supervision of Philipp Furtwängler...
who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem.