A group (G, *) has 10 elements. The minimum number of elements of G which are their own inverse is
Answers
Answer:
the answer is 2 and can be found in cyclic group Z_10
Explanation:
In a group G, with an even number of elements, other than the identity element 'e', which is of order 1, (and is its own inverse) there are an odd number of elements of order 2, which are their own inverse. This can be proved by pairing each elements with its inverse element.
A group of order 10, can be either isomorphic to the cyclic group Z_10, or the dihedral group D_10 of symmetries of a regular pentagon. The group Z_10 has the minimum number of self-inverse elements, viz. [0] and [5].
The dihedral group above has 5 rotations about the centre of the pentagon by integer multiples of 2π/5, and this includes 'e', and 5 reflections about the five medians corresponding to the five vertices joined with the respective midpoints of their opposite sides and these are all self-inverses. Hence there are 6 self-inverse elements in all, counting e.