Chapters four through seven cover abstract groups and monoids, orthogonal groups, stochastic matrices,

Lagrange's theorem, groups of units of monoids, homomorphisms, rings, and integral domains.

Among the topics are greatest common divisors, integer multiples and exponents, quotients of polynomial rings, divisibility and factorization in integral domains, subgroups of cyclic groups, cosets and

Lagrange's theorem, the fundamental theorem of finite abelian groups, and check digits.

From here, through

Lagrange's theorem, we calculate the top of the hyperbola of beryllium: X = 60.9097, Y = 0.14796.

First note that by

Lagrange's theorem g/h is an integer.

Lagrange's theorem states that for any subgroup K of a group H, H [congruent to] K x Q as (left) K-sets, where Q = H/K.

The main result of this paper (Theorem 7) is a version of Lagrange's theorem for Hopf monoids in the category of connected species.

Consequently, by

Lagrange's theorem, [intersection] = order([L.sup.1]) divides m!, and therefore m = n.

Note that by Lagrange's theorem, [absolute value of [S.sub.m]] divides [absolute value of [G.sub.n]].

We state without proof the famous Lagrange's Theorem:

It isn't hard for students to conjecture

Lagrange's theorem, which says that the order of a subgroup of any finite group must divide the order of the group.

We chose the top of the hyperbolas, in order to describe a chemical process with use of

Lagrange's theorem; reducing them to the equation Y = K/X was made through the scaling coefficient 20.2895, as we have deduced.

Lagrange's theorem is not satisfied by all subloops of the loop [L.sub.n](m),i.e there always exists a subloop H of [L.sub.n](m) which does not satisfy the

Lagrange's theorem, i.e o(H) [dagger] o([L.sub.n](m)).