LOG#085. Group theory(V).


Other important concepts and definitions in group theory!

Definition (22). Normal or invariant group. Let H be a subgroup of other group G. We say that H is a normal or invariant subgroup of G if the following condition holds:

H=xHx^{-1} \forall x\in G

Proposition. Let H be a subgroup of G. l H is invariant if and only if (iff) H is a union of conjugacy classes of G.

Check: H is normal iff a\in H\Rightarrow xax^{-1}\in H \forall x\in G. Thus, H is normal iff whenever it contains an element a, then it also contains the conjugacy class of a, 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 \left\{ e\right\}.

Definition (24). Semisimple group. We say that G is a semisimple group if there is no any “abelian invariant subgroup” except the trivial element \left\{ e\right\}.

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 H\subseteq G be an invariant subgroup. Then we say that

i) The set xH=\left\{xh/h\in G\right\} with x\in G is a coset by the left, and

ii) The set Hx=\left\{hx/h\in G\right\} sith x \in H is a coset by the right.

Theorem. The set A=\left\{ g_iH/g_i\in G\right\}

with composition law g_iH\circ g_jH\equiv (g_i\circ g_j)H is a group.

Definition (25). Quotient group. The group defined in the previous theorem is called quotient group and it is generally denoted by G/H\equiv (A,\circ).

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 (G,\circ) and (G',\circ ') be two different groups. Any map/application/function/functor

f: G\longrightarrow G'

is called an homomorphism if it preserve the operations of the respective groups (their “products” or “multiplications”) in the following sense

f(x\circ y)=f(x)\circ ' f(y) \forall x,y\in G

Definition (27). (Group) Isomorphism. Let f be an homomorphism, then f is an isomorphism if f is “bijective”, or one-to-one correspondence between the elements of G and G'.

Definition (28). Kernel. Let (G,\circ) and (G',\circ ') be two different groups. and f:G\longrightarrow G' a function between them. Then, we define the kernel of f as the following set

\mbox{Ker}(f)=\left\{g \in G/f(g)=e'\right\}

and where e' is the neutral element in the group G'.

There are two important theorems for group homomorphisms:

Theorem. First theorem of group homomorphisms. Let (G,\circ) and (G',\circ ') be two different groups. If we hav a function f:G\longrightarrow G' being a group homomorphism, then its kernel is an invariant subgroup of (G,\circ).

Property. It is clear from the above theorem that, if f is bijective, then we have the special case in which \mbox{Ker}(f)=e. Then, the theorem is trivially satisfied!

Theorem. Second theorem of group homomorphisms. Let (G,\circ) and (G',\circ ') be two different groups. If f:G\longrightarrow G' is a group homomorphism, then an aplication \iota exists such as

\iota= G/\mbox{Ker}(f)\longrightarrow \mbox{Im}(f)

such as \iota is a group isomorphism.

May the group theory be with you until the next blog post!

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