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5.8. SOME FACTS FROM GROUP THEORY 227

THEOREM. The homomorphic image f(G) is a group. The image f(e) of the identity
element e G is the identity element of the group f(G). Mutually inverse elements of G
correspond to mutually inverse images in f(G).

Two groups G 1 and G 2 are said to be isomorphic if there exists a one-to-one mapping
f of G 1 onto G 2 such that f(ab)= f(a)f(b)for all a, b G 1 . Such a mapping is called an
isomorphism or isomorphic mapping of the group G 1 onto the group G 2 .
THEOREM. Any isomorphism of groups is invertible, and the inverse mapping is also
an isomorphism.
An isomorphic mapping of a group G onto itself is called an automorphism of G.If
f 1 : G → G and f 2 : G → G are two automorphisms of a group G, one can define another
automorphism f 1 ◦ f 2 : G → G by letting (f 1 ◦ f 2 )(g)= f 1 (f 2 (g)) for all g G.This
automorphism is called the composition of f 1 and f 2 , and with this composition law, the set
of all automorphisms of G becomes a group called the automorphism group of G.

5.8.1-4. Subgroups. Cosets. Normal subgroups.

Let G be a group. A subset G 1 of the group G is called a subgroup if the following
conditions hold:
1. For any a and b belonging to G 1 , the product ab belongs to G 1 .
2. For any a belonging to G 1 , its inverse a –1 belongs to G 1 .
These conditions ensure that any subgroup of a group is itself a group.
Example 5. The identity element of a group is a subgroup. The subset of all even numbers is a subgroup
of the additive group of all integers.
The product of two subsets H 1 and H 2 of a group G is a set H 3 that consists of all
elements of the form h 1 h 2 ,where h 1 H 1 , h 2 H 2 . In this case, one writes H 3 = H 1 H 2 .
Let H be a subgroup of a group G and a some fixed element of G.The set aH is called
a left coset,and the set Ha is called a right coset of the subgroup H in G.
Properties of left cosets (right cosets have similar properties):
1. If a H,then aH ≡ H.
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2. Cosets aH and bH coincide if a b H.
3. Two cosets of the same subgroup H either coincide or have no common elements.
4. If aH is a coset, then a aH.
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A subgroup H of a group G is called a normal subgroup of G if H = a Ha for any
a G. This is equivalent to the condition that aH = Ha for any a G, i.e., every right
coset is a left coset.

5.8.1-5. Factor groups.

Let H be a normal subgroup of a group G. Then the product of two cosets aH and bH
(as subsets of G)isthe coset abH. Consider the set Q whose elements are cosets of the
subgroup H in G, and define the product of the elements of Q as the product of cosets.
Endowed with this product, Q becomes a group, denoted by Q = G/H and called the
quotient group of G with respect to the normal subgroup H.
The mapping f : G → G/H that maps each a G to the corresponding coset aH is a
homomorphism of G onto G/H.
If f : G → G is a homomorphism of groups, the set of all elements of G mapped into the
identity element of G is called the kernel of f and is denoted by ker f = {g G: f(g)= f(e)}.