find all homomorphic images of the octic group D4SolutionSup

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 = <r, s | r⁴ = s² = 1, rs = sr\'> (\' denoting the inverse). Through a tedious but standard calculation, the normal subgroups of D4 are {1}, <r²>, <s, r²>, <r>, <rs, r²>, and D4 itself.

D4/{1} is of course just D4.

<r²> here has order 2, so D4/<r²> 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 <r²>. D4/<r²> 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.