In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Schreier refinement theorem of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
states that any two
subnormal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...
of
subgroup
In group theory, a branch of mathematics, 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 ...
s of a given group have equivalent refinements, where two series are equivalent if there is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between their
factor group
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
s that sends each factor group to an
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
one.
The theorem is named after the
Austria
Austria, , bar, Östareich officially the Republic of Austria, is a country in the southern part of Central Europe, lying in the Eastern Alps. It is a federation of nine states, one of which is the capital, Vienna, the most populous ...
n
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Otto Schreier
Otto Schreier (3 March 1901 in Vienna, Austria – 2 June 1929 in Hamburg, Germany) was a Jewish-Austrian mathematician who made major contributions in combinatorial group theory and in the topology of Lie groups.
Life
His parents were the arc ...
who proved it in 1928. It provides an elegant proof of the
Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
. It is often proved using the
Zassenhaus lemma Zassenhaus is a German surname. Notable people with the surname include:
* Hans Zassenhaus (1912–1991), German mathematician
** Zassenhaus algorithm
** Zassenhaus group
** Zassenhaus lemma
* Hiltgunt Zassenhaus (1916–2004), German philologi ...
. gives a short proof by intersecting the terms in one subnormal series with those in the other series.
Example
Consider
, where
is the
symmetric group of degree 3
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3, or, in other words, the dihedral group of order 6. It is isomorphic to the symmetric group S3 of degree 3. It is also the smallest possible non-abeli ...
. The
alternating group
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or
Basic prop ...
is a normal subgroup of
, so we have the two subnormal series
:
:
with respective factor groups
and
.
The two subnormal series are not equivalent, but they have equivalent refinements:
:
with factor groups isomorphic to
and
:
with factor groups isomorphic to
.
References
*
Theorems in group theory
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