group axiom in a sentence
1) Thus a group homomorphism respects all the structure of "G" provided by the group axioms.
2) With the proper generalization of the group axioms this gives rise to an "n"-ary group.
3) The following properties of integer addition serve as a model for the abstract group axioms given in the definition below.
4) The only group axiom that requires some effort to verify is that each element of "G" is invertible.
group axiom example sentences5) Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements.
6) The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.
7) This means that we have a smooth multiplication μ : G × G → G, an inversion ν : G → G and a unit element e ∈ G such that the usual group axioms are satisfied.
8) It can be shown that all other relations among the braids __FORMULA__ and __FORMULA__ already follow from these relations and the group axioms.
9) In the case above, the inverse of 4 is 4, and the inverse of 3 is 2, as . Hence all group axioms are fulfilled.
10) Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under "elementary group theory".
11) Indeed the four group axioms are satisfied:
12) and thus the transformation matrix, consistent with the group axioms, is given by
13) Given this set of symmetries and the described operation, the group axioms can be understood as follows:
14) To qualify as a group, the set and operation,, must satisfy four requirements known as the " group axioms":
example sentences with axiom15) To qualify as an abelian group, the set and operation,, must satisfy five requirements known as the "abelian group axioms":
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