I have posted a revised version of the abstractmath.org article on mathematical definitions
here. I previously attempted to post it on this blog but the addition of the necessary css file had various side effects elsewhere in my Word Press setup, so I deleted the post.
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Just a quick correction: the monster group is the largest sporadic finite simple group, not the largest finite simple group. After all, there are arbitrarily large primes, and all cyclic groups of prime order are trivially finite and simple.