torsion subgroup

torsion subgroup

English

Noun

torsion subgroup (plural torsion subgroups)

1. (algebra) Subgroup of an abelian group whose generators have finite order.
   • 2006 July 7, Joseph H. Silverman, An Introduction to the Theory of Elliptic Curves‎:

     A standard theorem about finitely generated abelian groups tells us that ${\\displaystyle E(\\mathbb {Q} )}$ looks like
     ${\\displaystyle \\qquad E(\\mathbb {Q} )\\cong ({\\mbox{Finite Group}})\\times \\underbrace {\\mathbb {Z} \\times \\mathbb {Z} \\times ...\\times \\mathbb {Z} } _{r{\\mbox{ copies}}}}$
     ${\\displaystyle \\qquad E(\\mathbb {Q} )\\cong E(\\mathbb {Q} )_{\\mbox{tors}}\\times \\underbrace {\\mathbb {Z} \\times \\mathbb {Z} \\times ...\\times \\mathbb {Z} } _{r{\\mbox{ copies}}}}$
     The finite subgroup ${\\displaystyle E(\\mathbb {Q} )_{\\mbox{tors}}}$ is called the
     ${\\displaystyle \\qquad \\color {red}{\\mbox{Torsion Subgroup of }}E(\\mathbb {Q} ).}$