Assertion: abelian group's element | /r/HasAspect | order
Facets:
Source sentences:
text: The set {3,19} generates the group, which means that every element of ( Z / 20 Z ) × {\displaystyle (\mathbb {Z} /20\mathbb {Z} )^{\times }} is of the form 3a × 19b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
text: The set {3,19} generates the group, which means that every element of ( Z / 20 Z ) × {\displaystyle (\mathbb {Z} /20\mathbb {Z} )^{\times }} is of the form 3a × 19b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2).
text: an = e, we say that the element a has order 0.
text: All non-identity elements of the Klein group have order 2, thus any two non-identity elements can serve as generators in the above presentation.
text: Hence we can apply the first case and we conclude that the element $a’b’$ has order
text: The elements both have order either or .
text: We first prove the special case that where G is abelian, and then the general case; both proofs are by induction on n = |G|, and have as starting case n = p which is trivial because any non-identity element now has order
text: Therefore, no element of has order greater than 12.
text: Therefore, no element of has order greater than 12.