  In this paper, we study the symmetry of polar codes on symmetric binary-input
discrete memoryless channels (B-DMC). The symmetry property of polar codes is
originally pointed out in Arikan's work for general B-DMC channels. With the
symmetry, the output vector $y_1^N$ ($N$ be the block length) can be divided
into equivalence classes in terms of their transition probabilities. In this
paper, we present a new frame of analysis on the symmetry of polar codes for
B-DMC channels. Theorems are provided to characterize the symmetries among the
received vectors.
