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How Order of element is equal to its inverse in group?
The order of an elementg in a group is the least positive integer k such that gk is the identity.Now look at the same group, we know there exists an element h such that gh=hg=e where e is the identity. This must be true because existence of inverses is one of the conditions required for a set to be a group. So if gk=e and gh=e, then gk =gh and we see the relation between k, the order and h the inverse in the group.
What is the no of generator in n order group?
In abstract algebra, a generating set of a group is a subset of that group. In that subset, every element of the group can be expressed as the combination (under the group operation) of finitely many elements of the subset and their inverses.
How do you find proper subgroups of a cyclic group of order 6?
A cyclic group of order 6 is isomorphic to that generated by elements a and b where a2 = 1, b3 = 1, or to the group generated by c where c6 = 1. So, find the identity element, 1. Next find an element which when operated on by itself, equals the identity. This element will correspond to a or c3. Finally find an element which when operated on by itself twice (so that it is cubed or multiplied by 3), equals the identity. This element will correspond to b or c2. The subgroups {1}, (1, a} = {1, c3} and {1, b, b2} = {1, c2, c4} will be proper subgroups.
What is an identify propery?
In a group, the identity property is that each group contains an element, i, such that for all elements x, in the group, i*x = x*i = x. i is called the identity element.
What is the smallest subdivision of a group or set?
An element of that set.
