Trivial group

Summary

In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: or depending on the context. If the group operation is denoted then it is defined by The similarly defined is also a group since its only element is its own inverse, and is hence the same as the trivial group. The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group. Given any group the group consisting of only the identity element is a subgroup of and, being the trivial group, is called the of The term, when referred to " has no nontrivial proper subgroups" refers to the only subgroups of being the trivial group and the group itself. The trivial group is cyclic of order ; as such it may be denoted or If the group operation is called addition, the trivial group is usually denoted by If the group operation is called multiplication then 1 can be a notation for the trivial group. Combining these leads to the trivial ring in which the addition and multiplication operations are identical and The trivial group serves as the zero object in the , meaning it is both an initial object and a terminal object.

Official source

https://en.wikipedia.org/wiki/Trivial_group

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Related publications (1)

Elliptic curves over a finite field and the trace formula

Ian Nicholas Petrow

We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2(Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2(Z) which include as special cases the groups 1(N) and (N). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z)2 and congruence subgroups 0(2),0(4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vladu, Gekeler, Howe and others.

Wiley2017

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