Group (mathematics) - Elementary Consequences of The Group Axioms

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

abc = (ab) • c = a • (bc)

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:

    As if paralyzed by the national fear of ideas, the democratic distrust of whatever strikes beneath the prevailing platitudes, it evades all resolute and honest dealing with what, after all, must be every healthy literature’s elementary materials.
    —H.L. (Henry Lewis)

    We are still barely conscious of how harmful it is to treat children in a degrading manner. Treating them with respect and recognizing the consequences of their being humiliated are by no means intellectual matters; otherwise, their importance would long since have been generally recognized.
    Alice Miller (20th century)

    The poet who speaks out of the deepest instincts of man will be heard. The poet who creates a myth beyond the power of man to realize is gagged at the peril of the group that binds him. He is the true revolutionary: he builds a new world.
    Babette Deutsch (1895–1982)

    “I tell you the solemn truth that the doctrine of the Trinity is not so difficult to accept for a working proposition as any one of the axioms of physics.”
    Henry Brooks Adams (1838–1918)