You basically just write it down carefully. It is a matter of notation rather than a matter of mathematics. It is constructed to fulfill the universal property.Heidi said:
Why don't you try what I have said? Write down all morphisms that you have, name the group and the index set, and note what must be shown. I see no sense in rewriting the definition. So in order to communicate, we need a common language, not just M*N. Have you read the nLab link?Heidi said:
Ok, this simply means ##H=\{1\}.## The elements of ##G=G_1\ast G_2## are words over the alphabet of elements from ##G_1## and ##G_2.## Say we have ##abcxyacbza\in G## with ##a,b,c\in G_1## and ##x,y,z\in G_2.## Then we have a homomorphism ##f_1\, : \,G_1\longrightarrow Q## that maps ##a,b,c,## say onto ##\overline{a},\overline{b},\overline{c}## and the same for ##f_2\, : \,G_2\longrightarrow Q## that maps ##x,y,z## onto ##x',y',z'.## The dashed group homomorphism ##f## maps then ##abcxyacbza## onto ##f_1(a)f_1(b)f_1(c)f_2(x)f_2(y)f_1(a)f_1(c)f_1(b)f_2(z)f_1(a)= \overline{a}\overline{b}\overline{c}x'y'\overline{a}\overline{c}\overline{b}z'\overline{a}.##Heidi said: