i just don't really get the one step subgroup test, which is very important, and something i should understand. can someone walk me through in general how to use the test? maybe give me a simple example? thanks.
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Here's a simple example. Let G be the integers with addition as the operation. Let H be the subset of G consisting of the multiples of 3. In G, the [additive] inverse of x is -x.
To show H is a subgroup by that test you must show that if x and y are in H, then x + (-y) is in H.
x in H says x = 3m for some m in G and y in H says y = 3n for some n in G. The inverse of y is then -3n so the expression x + (-y) is just
3m + (-3n) = 3(m + (-n)), using the usual rules. This is a multiple of 3 and therefore in H. So H is a subgroup.