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We study group theoretical structures of the mKdV equation. The Schwarziantype mKdV equation has the global M\"{o}bius group symmetry. The Miuratransformation makes a connection between the mKdV equation and the KdVequation. We find the special local M\"{o}bius transformation on the mKdVone-soliton solution which can be regarded as the commutative KdV B\"{a}cklundtransformation can generate the mKdV cyclic symmetric $N$-soliton solution. Inthis algebraic construction to obtain multi-soliton solutions, we could observethe addition formula.

The Construction of the mKdV Cyclic Symmetric $N$-soliton Solution by the B\"{a}cklund Transformation