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Is this true?

If $G$ is a group of size $n$, and $X$ is a non-empty subset of $G$ then $X^n$ is a subgroup of $G$?

By $X^n$ I mean the set of all products of length $n$ from $X$.

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    Also posted to MathOverflow, http://mathoverflow.net/questions/109590/the-powers-of-non-empty-subset-of-a-group-that-generate-a-subgroup --- with no notification to either site. Not good.2012-10-14

1 Answers 1

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EDIT: please ignore this answer, which was based on not understanding the problem.

There's a standard way to check whether something is a subgroup.

Is it non-empty? Sure.

Is it closed under the operation? $(a_1a_2\cdots a_n)(b_1b_2\cdots b_n)=(a_1a_2)(a_3a_4)\cdots(b_{n-1}b_n)$ so that works.

Does it have the inverse of each of its elements? Sure, the inverse of a product of $n$ things is the product of the $n$ inverses (in the opposite order).

All done.

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    The argument is wrong. You implicitly used the fact that a product of two elements of $X$ and the inverse of an element of $X$ lies again in $X$. Thats it what you want to proove so you have a circle argument.2014-09-14