Let $\mathbb R$ be the set of all real numbers and $H = \mathbb R\times \mathbb R$. Define a binary operation $\circ$ on $H$ as follows $(p,q) \circ (r,s) = (pr,ps+q)$, where $(p,q),(r,s) \in H$? Is $(H, \circ )$ a group?
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To prove this is a group, you need to check all the axioms of a group. You need an identity, so can you find an element $(x,y)$ such that $(p,q)\circ (x,y)=(p,q)=(x,y)\circ(p,q)$. If you expand out the $\circ$'s it should be easy to figure out what $x$ and $y$ need to be. Then look for the inverse of a general element $(p,q)$. Then see if associativity is satisfied.
Hint: Find the identity of the group and show not all elements have inverse in the group.