Other important concepts and definitions in group theory!
Definition (22). Normal or invariant group. Let be a subgroup of other group G. We say that is a normal or invariant subgroup of G if the following condition holds:
Proposition. Let be a subgroup of G. is invariant if and only if (iff) is a union of conjugacy classes of .
Check: H is normal iff . Thus, H is normal iff whenever it contains an element , then it also contains the conjugacy class of , so another way to say this is that H is a union of conjugacy classes.
Definition (23). Simple group. We say that G is a simple group if there is no invariant subgroups except the trivial, the neutral/identity element .
Definition (24). Semisimple group. We say that G is a semisimple group if there is no any “abelian invariant subgroup” except the trivial element .
Simple groups are always semisimple, but the converse is not true. Simple and semisimple finite groups have been completely classified by mathematicians. We will talk about this later.
Definition (25). Coset. Let be an invariant subgroup. Then we say that
i) The set with is a coset by the left, and
ii) The set sith is a coset by the right.
Theorem. The set
with composition law is a group.
Definition (25). Quotient group. The group defined in the previous theorem is called quotient group and it is generally denoted by .
Remark: Informally speaking, the elements of the quotient group are “the difference” between the elements of G and those in H.
Now, some additional definitions about morphisms, homomorphisms and isomorphisms in group theory.
Definition (26). (Group) Homomorphism. Let and be two different groups. Any map/application/function/functor
is called an homomorphism if it preserve the operations of the respective groups (their “products” or “multiplications”) in the following sense
Definition (27). (Group) Isomorphism. Let be an homomorphism, then is an isomorphism if is “bijective”, or one-to-one correspondence between the elements of and .
Definition (28). Kernel. Let and be two different groups. and a function between them. Then, we define the kernel of f as the following set
and where is the neutral element in the group .
There are two important theorems for group homomorphisms:
Theorem. First theorem of group homomorphisms. Let and be two different groups. If we hav a function being a group homomorphism, then its kernel is an invariant subgroup of .
Property. It is clear from the above theorem that, if f is bijective, then we have the special case in which . Then, the theorem is trivially satisfied!
Theorem. Second theorem of group homomorphisms. Let and be two different groups. If is a group homomorphism, then an aplication exists such as
such as is a group isomorphism.
May the group theory be with you until the next blog post!