Abstract Algebra: Lagrange's Theorem 2

Hail Mathematicians,

My last question for the day... I think. Check my solution please.

__Problem:__

The *exponent* of a group is the smallest positive integer such that for each ( is the identity element), if such an integer exists. Prove that every finite group has an exponent, and that this exponent divides the order of the group. [Suggestion: Consider the LCM of .]

__Solution:__

We need to show: (1) If is finite, then has an exponent; and (2) we need to find such an exponent.

__Proof:__

(1) By the contrapositive, assume that does not have an exponent. Then there exists some such that for all . But that would mean is of infinite order (by the definition of ), and so, would have to be infinite.

(2) Consider . Now, for any , we have for some by (1). But since , we can write for some . Thus, . And since is the LCM of the orders of all , it is the smallest integer for which this is true. Moreover, as a corollary of Lagrange's Theorem.

QED

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