In mathematical
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb
Traditionally, a finite verb (from la, fīnītus, past partici ...
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 ...
, the Thompson order formula, introduced by
John Griggs Thompson
John Griggs Thompson (born October 13, 1932) is an American mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992, and the Abel Prize in 2008 ...
, gives a formula for the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
of a finite group in terms of the centralizers of involutions, extending the results of .
Statement
If a finite group ''G'' has exactly two conjugacy classes of involutions with representatives ''t'' and ''z'', then the Thompson order formula states
:
Here ''a''(''x'') is the number of pairs (''u'',''v'') with ''u'' conjugate to ''t'', ''v'' conjugate to ''z'', and ''x'' in the subgroup generated by ''uv''.
gives the following more complicated version of the Thompson order formula for the case when ''G'' has more than two conjugacy classes of involution.
:
where ''t'' and ''z'' are non-conjugate involutions, the sum is over a set of representatives ''x'' for the conjugacy classes of involutions, and ''a''(''x'') is the number of ordered pairs of involutions ''u'',''v'' such that ''u'' is conjugate to ''t'', ''v'' is conjugate to ''z'', and ''x'' is the involution in the subgroup generated by ''tz''.
Proof
The Thompson order formula can be rewritten as
:
where as before the sum is over a set of representatives ''x'' for the classes of involutions.
The left hand side is the number of pairs on involutions (''u'',''v'') with ''u'' conjugate to ''t'', ''v'' conjugate to ''z''. The right hand side counts these pairs in classes, depending the class of the involution in the cyclic group generated by ''uv''. The key point is that ''uv'' has even order (as if it had odd order then ''u'' and ''v'' would be conjugate) and so the group it generates contains a unique involution ''x''.
References
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*{{Citation , last1=Suzuki , first1=Michio , author1-link=Michio Suzuki (mathematician) , title=Group theory. II , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, location=Berlin, New York , series=Grundlehren der Mathematischen Wissenschaften
undamental Principles of Mathematical Sciences, isbn=978-0-387-10916-9 , mr=815926 , year=1986 , volume=248
Finite groups