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2. 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)

3. Symbols of grouping - Wikipedia

   en.wikipedia.org/wiki/Symbols_of_grouping

   [2] A superscript is understood to be grouped as long as it continues in the form of a superscript. For example if an x has a superscript of the forma+b, the sum is the exponent. For example: x 2+3, it is understood that the 2+3 is grouped, and that the exponent is the sum of 2 and 3. These rules are understood by all mathematicians.

4. Free group - Wikipedia

   en.wikipedia.org/wiki/Free_group

   Two free groups F S and F T are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every cardinal number k, there is, up to isomorphism, exactly one free group of rank k. A free group of finite rank n > 1 has an exponential growth rate of order 2n − 1. A few other ...

5. 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.

6. Product of groups - Wikipedia

   en.wikipedia.org/wiki/Product_of_groups

   Download as PDF; Printable version ... In mathematics, a product of groups usually refers to a direct product ... Product of group subsets; wreath product; free ...

7. 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 ...