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2. Symbols of grouping - Wikipedia

   en.wikipedia.org/wiki/Symbols_of_grouping

   If we were to express this idea using symbols of grouping, the factors in a product. Example: 2+3×4 = 2 +(3×4)=2+12=14. In understanding expressions without symbols of grouping, it is useful to think of subtraction as addition of the opposite, and to think of division as multiplication by the reciprocal.

3. Multiplicative group - Wikipedia

   en.wikipedia.org/wiki/Multiplicative_group

   The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a group scheme.That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product of schemes, with the morphism e that serves as the identity.

4. List of problems in loop theory and quasigroup theory

   en.wikipedia.org/wiki/List_of_problems_in_loop...

   Construct a conjugacy closed loop whose left multiplication group is not isomorphic to its right multiplication group. Proposed: by Aleš Drápal at Loops '03, Prague 2003; Solved by: Aleš Drápal; Solution: There is such a loop of order 9. In can be obtained in the LOOPS package by the command CCLoop(9,1)

5. Multiplicative order - Wikipedia

   en.wikipedia.org/wiki/Multiplicative_order

   The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Z n; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or ...

6. Free group - Wikipedia

   en.wikipedia.org/wiki/Free_group

   The Nielsen–Schreier theorem: Every subgroup of a free group is free. Furthermore, if the free group F has rank n and the subgroup H has index e in F, then H is free of rank 1 + e(n–1). A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all ...

7. Lamplighter group - Wikipedia

   en.wikipedia.org/wiki/Lamplighter_group

   The standard presentation for the lamplighter group arises from the wreath product structure ,, [,],, , which may be simplified to ,, (), . The growth rate of the group, the function describing the number of group elements that can be formed as a product of generators for each , is generally defined with respect to these two generators and .