4. For each of the subgroups {e, a 2 }and {e, b} of D 4 list all left and right

Question

4. For each of the subgroups {e, a2 }and {e, b} of D4 list all left and right cosets. If I am not wrong, I do believe the element of D4,are e, a a2,a3,b ab,a2b,a3b. So I multiplied each element of D4 time theelements of each of the subgroups ( {e, a2 } and{e, b}) say, let {e,a2} = H, and I will omitt the brackets forconvenience e*H = e, a2 a*H = a, a3 a2*H = a2,e a3*H = a3,a b*H = b,a2b ab*H = ab, a3b Now I have found all the element of D4 in six leftcosets. But the concern I have is, I though that each coset had toto have unique elements distinct from the other cosetsand I seerepeated elements in some cosets. And second I still have problems in how to proced with theright cosets. Thanks. 4. For each of the subgroups {e, a2 }and {e, b} of D4 list all left and right cosets. If I am not wrong, I do believe the element of D4,are e, a a2,a3,b ab,a2b,a3b. So I multiplied each element of D4 time theelements of each of the subgroups ( {e, a2 } and{e, b}) say, let {e,a2} = H, and I will omitt the brackets forconvenience e*H = e, a2 a*H = a, a3 a2*H = a2,e a3*H = a3,a b*H = b,a2b ab*H = ab, a3b Now I have found all the element of D4 in six leftcosets. But the concern I have is, I though that each coset had toto have unique elements distinct from the other cosetsand I seerepeated elements in some cosets. And second I still have problems in how to proced with theright cosets. Thanks. a2*H = a2,e a3*H = a3,a b*H = b,a2b ab*H = ab, a3b Now I have found all the element of D4 in six leftcosets. But the concern I have is, I though that each coset had toto have unique elements distinct from the other cosetsand I seerepeated elements in some cosets. And second I still have problems in how to proced with theright cosets. Thanks.

Explanation / Answer

Question Details:
SEE MY COMMENTS IN CAPITALS
4. For each of the subgroups H={e, a2 }and G={e, b} of D4 list all left and right cosets. If I am not wrong, I do believe the element of D4,are e, a , a2,a3,b, ab,a2b,a3b...OK
YOU ARE PERFECTLY RIGHT....ONLY BEING MODEST , I THINK.
So I multiplied each element of D4 time theelements of each of the subgroups ( {e, a2 } and{e, b}) say, let {e,a2} = H, and I will omitt the brackets forconvenience...........OK e*H = e, a2 .............OK
a*H = a, a3..........................OK a2*H = a2,e.....OK a3*H = a3,a................OK b*H = b,Ba2............... ab*H = ab, abA2= AB,ABAA=AB,AA-1BA= AB,BA
A2B*H=A2B,A2BA2=A2B,ABA =
A3B*H=A3B,A3BA2=A3B,A2BA =
Now I have found all the element of D4 in six leftcosets.
YOU LEFT THE LAST 2 ELEMENTS
But the concern I have is, I though that each coset had to tohave unique elements distinct from the
other cosets and I see repeated elements in somecosets. And second I still have problems in how to proced with theright cosets.
PROCEED IN THE SAME WAY USING H*D4...IN THE ABOVE YOU DID D4*H
PUTTING A4=E;B2=E;BA=A-1B
F IT IS A NORMAL SUB GROUP , YOU GET BOTH SAME.

REPEAT SAME FOR THE OTHER SUB GROUP.Thanks.

a2*H = a2,e.....OK a3*H = a3,a................OK b*H = b,Ba2............... ab*H = ab, abA2= AB,ABAA=AB,AA-1BA= AB,BA
A2B*H=A2B,A2BA2=A2B,ABA =
A3B*H=A3B,A3BA2=A3B,A2BA =
Now I have found all the element of D4 in six leftcosets.
YOU LEFT THE LAST 2 ELEMENTS
But the concern I have is, I though that each coset had to tohave unique elements distinct from the
other cosets and I see repeated elements in somecosets. And second I still have problems in how to proced with theright cosets.
PROCEED IN THE SAME WAY USING H*D4...IN THE ABOVE YOU DID D4*H
PUTTING A4=E;B2=E;BA=A-1B
F IT IS A NORMAL SUB GROUP , YOU GET BOTH SAME.

REPEAT SAME FOR THE OTHER SUB GROUP.Thanks.

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