Let me begin with a definition.Originally Posted byTexasGirl

Definition:An abelian group is a "vector-space" over field . When there exists a binary operation:

such as,

.

Definition:If subset of a vector space is a vector space over the same field. It is called a "subspace".

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No, because we need that be a subgroup of but that is not true because are pairs, while are triples. Thus, these sets have totally differenet elements.Originally Posted byTexasGirl

Yes, because the binary operation we defined on over stasfies conditions 2,3,4,5. It only is questionable whether or not this binary operation is closed for over , we can see that it is closed. And finally is your trivial subgroup of thus, is aOriginally Posted byTexasGirlsubspaceofover.

No, because what happens if is irrational. Then, the binary operation is not closed because rationals multiplied by irrationals become irrationals.Originally Posted byTexaxGirl

Yes, again properties 2,3,4,5. Are true because they depend on the same binary operations that are on . Checking, whether it is closed. Notice when multiplied by a rational numbers becomes,Originally Posted byTexasGirl

notice that is still real and is still rational. Thus,

is asubspaceofover