An irreducible group is a group that cannot be factored into smaller groups. In other words, it does not have any non-trivial normal subgroups.
Irreducible groups are important in mathematics because they can help simplify complex equations and geometric figures. They also have applications in other fields such as physics and chemistry.
GL(n,F) is the general linear group, which consists of all invertible n-by-n matrices with entries from a field F. An irreducible group is a subgroup of GL(n,F) that cannot be further reduced, making it an important building block for studying the structure of GL(n,F).
No, not all groups are irreducible. Some groups can be broken down into smaller groups, making them reducible. However, irreducible groups play a significant role in mathematics and are often the focus of study.
Some examples of irreducible groups include cyclic groups, symmetric groups, and alternating groups. Other examples can be found in fields such as algebraic geometry and number theory.