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The properties of a subgroup would include the identity of the subgroup being the identity of the group and the inverse of an element of the subgroup would be the same in the group. The intersection of two subgroups would be a separate group in the system.
Closure, an identity element, inverse elements, associative property, commutative property
One is the multiplicative identity or the identity of/for multiplication.
It is the additive identity.
A guarantee provided by a corporation, a legal person, is known is corporate guarantee.