If H and K are subgroups of orders p and n , respectively, where p is a prime number, show that either H∩K={e} or H≤K .
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If G is a group and K≤H≤G , then show that [G:K]=[G:H][H:K] (assume finite indices). Hence deduce that if [G:H]=p, where p is a prime, then either H=K or H=G .
If G is a group and H,K are two subgroups of finite indices in G, prove that H∩K is of finite index in G. Can you find an upper bound for the index of H∩K in G?
If H and K are subgroups of finite indices of a group G such that [G:H] and [G:K] are relatively prime, show that G=HK .
For any subgroup H of G and any non-empty subset A of G define NH(A)={h∈H|hAh−1=A} . Show that NH(A)=NG(A)∩H and dedue that NH(A)≤H .
Let h≤G . Let N=⋂x∈GxHx−1 . Prove that N≤G, and aNa−1=N for all a∈G .
If H is a subgroup of finite index in G , prove that there is only a finite number of distinct subgroups in G of the form aHa−1 .
If H is of finite index in G, prove that there is a subgroup N of G, contained in H and of finite index in G such that aNa−1=N for all a∈G .
Let G be a finite group. Show that the only homomorphism from G to (Z,+)( or to (Q,+) or to (R,+)) is the zero homomorphism.
Show that any endomorphism of (Q,+) is multiplication by a rational number.
Prove that there is no non-trivial homomorphism from
(i) (Q,+) to (Z,+)
(ii) (Q,+) to (Zm,+)
(iii) (Q,+)to (Q∗,×), where Q∗=Q−{0}.
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