- #1
Bashyboy
- 1,421
- 5
Homework Statement
Let ##G##, ##H##, and ##K## be groups with homomorphisms ##\sigma_1 : K \rightarrow G## and ##\sigma_2 : K \rightarrow H##. Does there exist a homomorphism ##f: K \rightarrow G \times H## such that ##\pi_G \circ f = \sigma_1## and ##\pi_H \circ f = \sigma_2##? Is this function unique?
If either ##\sigma_1## or ##\sigma_2## are monomorphisms, then ##f## will also be a monomorphism.
Homework Equations
The Attempt at a Solution
Define ##f## to be the mapping ##f : K \rightarrow G \times H##.
##\pi_G ((g,h)) = g## and ##\pi_H((g,h)) = h##, where ##(g,h) \in G \times H##.
##\pi_G \circ f = \pi_G (f(k))##, where ##k \in K##.
##\pi_G \circ f = \pi_G(f(k)) = \pi((g,h)) = g \in G##
There exists an ##k_1 \in K## such that ##\sigma_1 (k_1) = g##. Thus,
##\pi_G \circ f = \sigma_1(k_1)## or
##\pi_G \circ f = \sigma_1##.
I did something similar for ##\pi_H \circ f##. However, this feels unsettling. Also, how would I even verify that ##f## is a homomorphism, if I do not know the rule of its mapping? Would I have to contrive a rule?