ehrenfest said:Homework Statement
I am pretty sure it is true since ax=b always has a solution if a and b are in group. But can someone just confirm this?
EDIT: sorry, I don't mean isomorphism, I mean bijection
Group theory is a branch of mathematics that studies the properties of groups, which are mathematical structures consisting of a set of elements and an operation that combines any two elements to produce a third element. Groups are used to describe and analyze various types of symmetry, as well as to study patterns and relationships in mathematics, physics, and other fields.
In group theory, an isomorphism is a type of function that preserves the structure and operations of a group. This means that an isomorphism between two groups will map elements from one group to corresponding elements in the other group while preserving the group's operation and identity element.
In a group, left multiplication by an element g means taking each element in the group and performing the group operation with g as the first element. So, if we have a group G and an element g in G, left multiplication by g would involve multiplying g with every other element in G.
No, left multiplication by g is only an isomorphism in a group if g is an element in the group's center. The center of a group is the set of elements that commute with all other elements in the group. If g is not in the center, then left multiplication by g will not preserve the group's structure and operations, and therefore, is not an isomorphism.
Studying left multiplication by g in a group can provide valuable insights into the structure of the group. It can help identify the center of the group, which is an important subgroup with many useful properties. Additionally, understanding left multiplication by g can also help in determining whether g is an automorphism, a special type of isomorphism that maps a group to itself.