Subject GRE-Style: Order of an Element

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Order of an Element 3 worked problems on finding element orders in groups with GRE speed techniques Key Recall: The order of an element in a group is the smallest positive integer such that . If no such exists, has infinite order. Question: What is the order of in ? (A) 4 (B) 5 (C) 10 (D) 20 (E) 8 Try it yourself first, then click to see the solution. Speed Technique: Use the Formula GRE Shortcut: In , the order of is . Verification: , and no smaller multiple of 8 is divisible by 20. Question: What is the order of the permutation in ? (A) 9 (B) 12 (C) 24 (D) 6 (E) 36 Think about what determines the order of a product of disjoint cycles. Key Fact: The order of a product of disjoint cycles equals the LCM of the cycle lengths. Why? After 12 applications, each cycle returns to the identity: , , and . No smaller number is divisible by all three. (A) 6 (B) 10 (C) 15 (D) 30 (E) 60 How do orders in direct products relate to orders in each factor? Key Fact: In , the order of is . GRE Tip: Always compute individual orders first using , then take the LCM. This three-step process takes under 30 seconds.

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