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!