find all homomorphic images of the octic group D4 Solution Suppose f:D4 -> G is a homomorphism. ker(f) is a normal subgroup of D4, and from the first isomorphism theorem, D4/ker(f) = f(D4). That is, the homomorphic images are just the quotient groups (in general). We use the presentation D4 = (\' denoting the inverse). Through a tedious but standard calculation, the normal subgroups of D4 are {1}, , , , , and D4 itself. D4/{1} is of course just D4. here has order 2, so D4/ has order 4, and is thus either the Klein 4-group (Viergruppe) or the cyclic group of order 4. Taking a quotient cannot increase the order of an element, and only r and r\' have order 4 in D4, yet their homomorphic images have order 2 since r2 and (r\')2 are both in . D4/ must then be the Viergruppe. The remaining proper normal subgroups all have order 4, so the quotient has order 8/4 = 2, and hence must be the cyclic group of order 2. D4/D4 = 1 obviously. An explicit homomorphism giving each of these images is the natural projection map induced by a given normal subgroup..

1. find all homomorphic images of the octic group D4
Solution
Suppose f:D4 -> G is a homomorphism. ker(f) is a normal subgroup of D4, and from the first
isomorphism theorem, D4/ker(f) = f(D4). That is, the homomorphic images are just the quotient
groups (in general).
We use the presentation D4 = (' denoting the inverse). Through a tedious but standard
calculation, the normal subgroups of D4 are {1}, , , , , and D4 itself.
D4/{1} is of course just D4.
here has order 2, so D4/ has order 4, and is thus either the Klein 4-group (Viergruppe) or the
cyclic group of order 4. Taking a quotient cannot increase the order of an element, and only r and
r' have order 4 in D4, yet their homomorphic images have order 2 since r2 and (r')2 are both in .
D4/ must then be the Viergruppe.
The remaining proper normal subgroups all have order 4, so the quotient has order 8/4 = 2, and
hence must be the cyclic group of order 2.
D4/D4 = 1 obviously.
An explicit homomorphism giving each of these images is the natural projection map induced
by a given normal subgroup.