1.6.45 Fields, Groups, Rings

Condition: a) the set \(G\)-set of real numbers is a group by addition; b) the set \( A \)-set of integers is a normal divisor in the group \( R \); c) describe the factor group \( G / A \), indicating in it the unit, the type of inverse element, and the group operation; d) prove that the mapping \( \varphi: H \rightarrow L \), where \( H- \) the group of integers that are multiples of 7 by addition, and \( L-Z^{28} \) the group of residues by mode 28, is a homomorphism, calculate its kernel and image. \( \varphi(x)=[x]_{28} \) - remainder of division by 28. e) construct a factor group \( H / \operatorname{Ker} \varphi \), indicating the multiplication table.