Question

In: Advanced Math

Prove that each element in Pentagon D5 has a unique inverse under the binary operation. D5={AF,...

Prove that each element in Pentagon D5 has a unique inverse under the binary operation.

D5={AF, BF, CF, DF, EF,0,72,144,216,288}

Expert Solution

Solution:

Set of symmetries of an regular pentagon with vertices levelled as in anticlockwise order. There are rotations ( ) where means rotation about degrees and reflections () where means reflection about the vertices .

Observe that

We can consider each rotations and reflections as a bijective function. For example in ,If we take an reflection say ,then
rotation about zero degree. this means there is no change in the original figure, so is treated as an identity function.
since we saw that each element of can be treated as a bijective function, so it is obvious that we can take binary operation as composition of function.
_{[This example shows you
how the operation works.]}

We can prove the statement through Cayley Table.

From the Cayley table one can say that the inverse element of that is , similarly and . Here from the Cayley table we conclude that each element of has unique inverse element. hence proved.

Colby Messinger answered 3 years ago

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