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Is there a set of any numbers, matrices or their generalizations that satisfies all of the following?

1) $A^2=B^2=C^2=D^2... =0$ where $A,B,C,D..$ are unequal mathematical objects

2) Objects in the set commute.

3) Other products of objects that do not involve the squared number of an object are not zero.

Also, what would be the restriction on the number of objects?

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    If you take any group that fulfills 1), it follows that 2) and 3) are true.2012-10-29
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    How? I do get 1) and 2), but not 3). Say, $ABC$. (group operation or multiplication symbol omitted.) Would this be nonzero?2012-10-29
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    If you want *all* products without repetition to be nonzero, you are asking too much --- if $AB=C$, then $ABC=0$. By the way, you've posted about a dozen questions now, under several aliases, all concerning the same circle of ideas --- just what is it that you are trying to do?2012-10-29
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    @Gerry Myerson. Nice observation.2012-10-29

2 Answers 2

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One example is the group $\mathbb{Z}_2 \times\mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$.

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In a group $G$, it can be easily shown that $1)\implies 2)$. But $2)\implies 3$ is wrong if the order of the group is odd. You can easily show that:

If $G$ is abelian of odd order, then the product of all elements of $G$ is the identity element. So now the case if $G$ is even is left.

Hence (abelian) groups of odd order do not work in your case.

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    Also if $|G|=$ even, then the product of elements is identity, which can be seen by using the same argument as in the odd case.2012-10-29
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    In the cyclic group of 4 elements, the product of the elements is not the identity.2012-10-29
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    Oh, so this does not work for groups of even order. Thanks.2012-10-29