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The order of an elementg in a group is the least positive integer k such that gk is the identity.
Now look at the same group, we know there exists an element h such that gh=hg=e where e is the identity. This must be true because existence of inverses is one of the conditions required for a set to be a group. So if gk=e and gh=e, then gk =gh and we see the relation between k, the order and h the inverse in the group.
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Why does every rational number have a additive inverse?
The rational numbers form an algebraic structure with respect to addition and this structure is called a group. And it is the property of a group that every element in it has an additive inverse.
What is the order of an element in a group?
The order of an element in a multiplicative group is the power to which it must be raised to get the identity element.
What is the additive inverse of 9.1?
We are talking group theory here. A group with addition has an additive inverse. A group with multiplication has a multiplicative inverse. The additive inverse of a number x is a y with x + y = 0. The additive inverse of x is written -x. Hence, the additive inverse of 9.1 equals -9.1. The reason that this question can arise is that beyond groups, there are rings and fields. Rings and fields have, besides addition, also multiplication. An element can have an additive inverse and a multiplicative inverse at the same time.
What are the properties of mathematical system to be a commutative group?
Closure, an identity element, inverse elements, associative property, commutative property
What are the properties of a subgroup?
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.
