440 Theorems

Question 1

1)
a finite group G is a p-group iff |G| is a power of p

2)
The center of a group G non-trivial p-group contains more than 1 element

Answer 1

1)
a finite group G is a p-group iff |G| is a power of p

2)
The center of a group G non-trivial p-group contains more than 1 element

Question 2

If H isap-subgroup of a finite group G, then [N_(H): H] ≡ G: H.

Answer 2

If H isap-subgroup of a finite group G, then [N_(H): H] ≡ G: H.

Question 3

If N is a normal subgroup of a group G, then every subgroup of G/N is of the form K/N, where K is a subgroup of G that contains N. Furthermore, K/N is normal in G/N if and only if K is normal in G.

Answer 3

If N is a normal subgroup of a group G, then every subgroup of G/N is of the form K/N, where K is a subgroup of G that contains N. Furthermore, K/N is normal in G/N if and only if K is normal in G.

Question 4

If f : G → H is an epimorphism of groups, then the assignment K → f(K) defines a one-to-one correspondence between the set S_f(G) of all subgroups K of G which contain Ker f and and set S(H) of all subgroups of H. Under this correspondence normal subgroups correspond to normal subgroups.

Answer 4

If f : G → H is an epimorphism of groups, then the assignment K → f(K) defines a one-to-one correspondence between the set S_f(G) of all subgroups K of G which contain Ker f and and set S(H) of all subgroups of H. Under this correspondence normal subgroups correspond to normal subgroups.

Question 5

If H is a p-subgroup of a finite group G such that p divides [G: H], then N_G(H)= H.

Answer 5

If H is a p-subgroup of a finite group G such that p divides [G: H], then N_G(H)= H.