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So, i was wondering if i wanted to find the total number of automorphisms of D_4(octic group) then what would that number be?

My answer is 4. Here, is how i reasoned about it. SInce D_4 is generated by only two elements (1234) and (24) i assume it is sufficient to fix their images, once we have done this, we have well defined one automorphism. NOw, we know that the isomorphism preserves the order of each element. THis means that the images of (1234) can be only (1234) itself and (1432). So, all isomorphisms (automorphisms) would start like this:

(1234)-->(1234) and (24)-->(24)

(1234)-->(1234) and (24)-->(13)

(1234)-->(1432) and (24)-->(24)

(1234)-->(1432) and (24)-->(13)

So, we would end up having only 4 automorphisms. Is this correct?

Edit: I just realized that there should be 8 automorphisms, but how do i go about finding the other 4?

My answer is 4. Here, is how i reasoned about it. SInce D_4 is generated by only two elements (1234) and (24) i assume it is sufficient to fix their images, once we have done this, we have well defined one automorphism. NOw, we know that the isomorphism preserves the order of each element. THis means that the images of (1234) can be only (1234) itself and (1432). So, all isomorphisms (automorphisms) would start like this:

(1234)-->(1234) and (24)-->(24)

(1234)-->(1234) and (24)-->(13)

(1234)-->(1432) and (24)-->(24)

(1234)-->(1432) and (24)-->(13)

So, we would end up having only 4 automorphisms. Is this correct?

Edit: I just realized that there should be 8 automorphisms, but how do i go about finding the other 4?

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