Elementary Consequences of The Group Axioms
Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of
- a • b • c = (a • b) • c = a • (b • c)
generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.
The axioms may be weakened to assert only the existence of a left identity and left inverses. Both can be shown to be actually two-sided, so the resulting definition is equivalent to the one given above.
Read more about this topic: Group (mathematics)
Famous quotes containing the words elementary, consequences, group and/or axioms:
“When the Devil quotes Scriptures, it’s not, really, to deceive, but simply that the masses are so ignorant of theology that somebody has to teach them the elementary texts before he can seduce them.”
—Paul Goodman (1911–1972)
“There are more consequences to a shipwreck than the underwriters notice.”
—Henry David Thoreau (1817–1862)
“No group and no government can properly prescribe precisely what should constitute the body of knowledge with which true education is concerned.”
—Franklin D. Roosevelt (1882–1945)
“The axioms of physics translate the laws of ethics. Thus, “the whole is greater than its part;” “reaction is equal to action;” “the smallest weight may be made to lift the greatest, the difference of weight being compensated by time;” and many the like propositions, which have an ethical as well as physical sense. These propositions have a much more extensive and universal sense when applied to human life, than when confined to technical use.”
—Ralph Waldo Emerson (1803–1882)