1 thought on “What does this definition of exchange group mean”
Roderick
Exchange group (Abel group)
definition 16.10 The operation "*" in the group u003Cg,*> is an exchanging operation, which is called the group u003Cg,*> is an exchange group ( Group) (Abel group).
example 16.18 group u003Cz, >, u003Cr, >, u003Cq, >, u003Cc, > all exchanges group.
Theorem 16.16 Set u003Cg,*> is a group, then u003Cg,*> The sufficient necessary conditions for the exchange group are:
There are (a*b) 2 = a2*b2
proved the necessity: for A, B∈g, because the operation "*" is exchanged, so there is
(a*b) 2 = (a*b)*(a*b) = a*(b*a)*b
= a*(a*b)*b = (a* a)*(b*b) = a2*b2
P everyone and sufficient: for A, B∈g, if (a*b) 2 = A2*b2, then
(a*b)*(a*b) = (a*a)*(b*b)
a*(b*a)*b = a*(a*b)* B
This to know: B*a = a*b
So the operation "*" meets the exchange law, that is, group u003Cg,*> is the exchange group.
Exchange group (Abel group)
definition 16.10 The operation "*" in the group u003Cg,*> is an exchanging operation, which is called the group u003Cg,*> is an exchange group ( Group) (Abel group).
example 16.18 group u003Cz, >, u003Cr, >, u003Cq, >, u003Cc, > all exchanges group.
Theorem 16.16 Set u003Cg,*> is a group, then u003Cg,*> The sufficient necessary conditions for the exchange group are:
There are (a*b) 2 = a2*b2
proved the necessity: for A, B∈g, because the operation "*" is exchanged, so there is
(a*b) 2 = (a*b)*(a*b) = a*(b*a)*b
= a*(a*b)*b = (a* a)*(b*b) = a2*b2
P everyone and sufficient: for A, B∈g, if (a*b) 2 = A2*b2, then
(a*b)*(a*b) = (a*a)*(b*b)
a*(b*a)*b = a*(a*b)* B
This to know: B*a = a*b
So the operation "*" meets the exchange law, that is, group u003Cg,*> is the exchange group.