Since |a| = 2, aba = b² implies that b = ab²a. Squaring both sides gives b² = (ab²a)(ab²a) = ab⁴a. Thus, aba = ab⁴a, which means b = b⁴ (by cancellation), so e = b³. Thus, the order of b divides 3. Since b is a nonidentity element, this means |b| = 3.
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Since |a| = 2, aba = b² implies that b = ab²a. Squaring both sides gives b² = (ab²a)(ab²a) = ab⁴a. Thus, aba = ab⁴a, which means b = b⁴ (by cancellation), so e = b³. Thus, the order of b divides 3. Since b is a nonidentity element, this means |b| = 3.