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Iwasawa‐type Decomposition in Compact Groups
Author(s) -
ITZKOWITZ GERALD,
WU TA SUN
Publication year - 1996
Publication title -
annals of the new york academy of sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.712
H-Index - 248
eISSN - 1749-6632
pISSN - 0077-8923
DOI - 10.1111/j.1749-6632.1996.tb36805.x
Subject(s) - mathematics , subgroup , abelian group , group (periodic table) , locally compact space , center (category theory) , combinatorics , compact group , type (biology) , centralizer and normalizer , locally compact group , discrete mathematics , pure mathematics , normal subgroup , physics , lie group , chemistry , crystallography , ecology , quantum mechanics , biology
Iwasawa in his famous paper “On some types of topological groups,” proved a number of decomposition theorems for compact groups. Among them is the theorem stating that a compact connected group G can be written as G = Z ( G ) · [ G , G ], where Z ( G ) is the center of G and [ G , G ] is the commutator subgroup of G. Later authors have refined this result to the statement that G = Z ( G ) o · [ G , G ], where Z ( G ) o is the component of the identity of Z ( G ). In the present paper we show that these theorems may be further strenghthened to the following statement: Let G be a compact group and let N be a compact normal subgroup of G. If either G = A · N , where A is a compact connected Abelian group, or if G / N is connected, then G = Z G ( N ) o · N , where Z G ( N ) o is the component of the identity of the centralizer in G of N. The proof of this result has a number of interesting corollaries.