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2. 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 ...

3. 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)

4. 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.

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. Presentation of a group - Wikipedia

   en.wikipedia.org/wiki/Presentation_of_a_group

   To see this, given a group G, consider the free group F G on G. By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G, therefore is equal to its normal closure, so G | K = F G /K.

7. 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.