MathStructures https://math.chapman.edu/~jipsen/structures/ 2022-09-28T21:38:04+00:00 MathStructures https://math.chapman.edu/~jipsen/structures/ https://math.chapman.edu/~jipsen/structures/lib/tpl/monobook/images/favicon.ico text/html 2021-05-04T17:24:04+00:00 jipsen (jipsen@undisclosed.example.com) start https://math.chapman.edu/~jipsen/structures/doku.php?id=start&rev=1620149044&do=diff Mathematical Structures The webpages collected here list information about classes of mathematical structures. The aim is to have a central place to check what properties are known about these structures. These pages are currently still under construction. text/html 2021-02-22T20:59:04+00:00 jipsen (jipsen@undisclosed.example.com) abelian_groups https://math.chapman.edu/~jipsen/structures/doku.php?id=abelian_groups&rev=1614027544&do=diff Abelian groups Abbreviation: AbGrp nbsp nbsp nbsp nbsp nbsp Abelian group Definition An \emph{abelian group} is a structure $\mathbf{G}=\langle G,+,-,0\rangle$, where $+$ is an infix binary operation, called the \emph{group addition}, $-$ is a prefix unary operation, called the \emph{group negative} and $0$$+$$x+y=y+x$$+$$(x+y)+z=x+(y+z)$$0$$+$$0+x=x$$-$$+$$-x+x=0$$\mathbf{G}$$\mathbf{H}$$\mathbf{G}$$\mathbf{H}$$h:G\rightarrow H$$h(x+y)=h(x)+h(y)$$h(-x)= -h(x)$$h(0)=0$\$\langle \mathbb{Z}, +…