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**Definition 3.7.1.**
Let G_{1} and G_{2} be groups, and let
µ : G_{1} -> G_{2}
be a function. Then µ
is said to be a
**group homomorphism** if

µ(ab) = µ(a) µ(b)

for all a,b in G

**Example** 3.7.1.
(Exponential functions for groups)
Let G be any group,
and let a be any element of G.
Define
µ : **Z** -> G
by µ(n) = a^{n},
for all n in **Z**.
This is a group homomorphism from **Z** to G.

If G is abelian, with its operation denoted additively,
then we define
µ : **Z** -> G
by µ(n) = na.

**Example** 3.7.2.
(Linear transformations)
Let V and W be vector spaces.
Since any vector space is an abelian group under vector addition,
any linear transformation between vector spaces is a group homomorphism.

**Proposition 3.7.2.**
If µ : G_{1} -> G_{2}
is a group homomorphism, then

**(a)**µ(e) = e;**(b)****(**µ(a)**)**^{-1}= µ(a^{-1}) for all a in G_{1};**(c)**for any integer n and any a in G_{1}, we have µ(a^{n}) =**(**µ(a)**)**^{n};**(d)**if a is in G_{1}and a has order n, then the order of µ(a) in G_{2}is a divisor of n.

**Example** 3.7.5.
(Homomorphisms from
**Z**_{n}
to **Z**_{k})
Any homomorphism
µ
: **Z**_{n} -> **Z**_{k}
is completely determined by
µ([1]_{n}),
and this must be an element
[m]_{k}
of **Z**_{k}
whose order is a divisor of n.
Then the formula
µ([x]_{n}) = [mx]_{k},
for all [x]_{n} in **Z**_{n}, defines a homomorphism.
Furthermore, every homomorphism from
**Z**_{n}
into **Z**_{k}
must be of this form. The image
µ(**Z**_{n})
is the cyclic subgroup generated by
[m]_{k}.

**Definition 3.7.3.**
Let µ : G_{1} -> G_{2}
be a group homomorphism. Then

{ x in G_{1} |
µ(x) = e }

**Proposition 3.7.4.**
Let µ : G_{1} -> G_{2}
be a group homomorphism, with
K = ker(µ).

**(a)**K is a normal subgroup of G.**(b)**The homomorphism µ is one-to-one if and only if K = {e}.

**Definition 3.7.5.**
A subgroup H of the group G is called a
**normal**
subgroup if

ghg^{-1} belongs to H

**Proposition 3.7.6.**
Let
µ : G_{1} -> G_{2}
be a group homomorphism.

**(a)**If H_{1}is a subgroup of G_{1}, then µ(H_{1}) is a subgroup of G_{2}.

If µ is onto and H_{1}is normal in G_{1}, then µ(H_{1}) is normal in G_{2}.**(b)**If H_{2}is a subgroup of G_{2}, thenµ

is a subgroup of G^{-1}(H_{2}) = { x in G_{1}| µ(x) is in H_{2}}_{1}.

If H_{2}is normal in G_{2}, then µ^{-1}(H_{2}) is normal in G_{1}.

Let µ : G_{1} -> G_{2}
be a group homomorphism.
The function µ
determines an equivalence relation of G_{1} by setting
a **~** b if
µ(a) =
µ(b).
The notation G_{1}/µ
is used for the set of equivalence classes of this relation.

**Proposition 3.7.7.**
Let µ
: G_{1} -> G_{2}
be a group homomorphism.
Then multiplication of equivalence classes in
G_{1}/µ
is well-defined, and
G_{1}/µ
is a group under this multiplication. The natural mapping
: G_{1} -> G_{1}/µ
defined by
(x)
= [x]_{µ}
is a group homomorphism.

**Theorem 3.7.8.**
Let µ : G_{1} -> G_{2}
be a group homomorphism.
Then G_{1}/µ
is isomorphic to µ(G_{1}).

**Proposition 3.7.9.**
Let µ : G_{1} -> G_{2}
be a group homomorphism, and let a,b belong to G_{1}.
The following conditions are equivalent:

**(1)**µ(a) = µ(b)**(2)**ab^{-1}belongs to ker(µ);**(3)**a = kb for some k in ker(µ);**(4)**b^{-1}a belongs to ker(µ);**(5)**a = bk for some k in ker(µ);

Theorem 3.7.8 is the most important result, but
Proposition 3.7.6 is also useful,
since for any group homomorphism µ : G_{1} -> G_{2}
it describes the connections between
subgroups of G_{1} and subgroups of G_{2}.
Examples 3.7.4 and
3.7.5 are important,
because they give a complete description of all
group homomorphisms between two cyclic groups.

**17.**
Find all group homomorphisms from
**Z**_{4} into **Z**_{10}.
*Solution*

**18.**
(a) Find the formulas for all group homomorphisms
from **Z**_{18} into **Z**_{30}.

(b) Choose one of the nonzero formulas in part (a),
and for this formula find the kernel and image,
and show how elements of the image
correspond to cosets of the kernel.

*Solution*

**19.**
(a) Show that **Z**_{7}^{×} is cyclic,
with generator [3]_{7}.

(b) Show that **Z**_{17}^{×} is cyclic,
with generator [3]_{17}.

(c) Completely determine all group homomorphisms from
**Z**_{17}^{×} into
**Z**_{7}^{×}.

*Solution*

**20.**
Define µ
: **Z**_{4} × **Z**_{6}
-> **Z**_{4} × **Z**_{3} by

µ ([x]_{4},[y]_{6})
= ([x+2y]_{4},[y]_{3}).

(b) Find the kernel and image of µ, and apply the fundamental homomorphism theorem.

**21.**
Let n and m be positive integers, such that m is a divisor of n. Show that

µ : **Z**_{n}^{×} ->
**Z**_{m}^{×} defined by

µ ( [x]_{n} ) = [x]_{m},
for all [x]_{n} in
**Z**_{n}^{×},

**22.**
For the group homomorphism
µ : **Z**_{36}^{×}
-> **Z**_{12}^{×} defined by

µ ( [x]_{36} ) = [x]_{12},
for all [x]_{36} in **Z**_{36}^{×},

**23.**
Let G, G_{1}, and G_{2} be groups.
Let µ_{1} : G -> G_{1}
and µ_{2} : G -> G_{2}
be group homomorphisms.
Prove that

µ : G -> G_{1} × G_{2}
defined by

µ (x) = (µ_{1} (x),
µ_{2} (x)),
for all x in G,

**24.**
Let p and q be different odd primes.
Prove that **Z**_{pq}^{×} is isomorphic to
the direct product
**Z**_{p}^{×} ×
**Z**_{q}^{×} .
*Solution*

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