Page 92 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
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3.2 Basic notions and examples
2. Let H be a subgroup of a group G with |G : H | = 2. Prove that H is a normal sub-
group of G .
3. Is it always true that if H is a subgroup of G with prime index, then H G ?
4. Let p be the smallest prime that divides the order of a finite group G . If H is a
subgroup of G of index p , then H is normal in G .
5. Find a group G and subgroups H and K with the property that H K G , but H is
not normal in G .
6. Let H and K be subgroups of finite index in G . Prove that |G : H ∩K | ≤ |G : H |·|G : K |,
with equality if and only if G = H K .
7. If H is a subgroup of G of finite index, then H contains a subgroup of finite index
which is normal in G .
8. A group in which every non-trivial element has order 2 is abelian.
9. Let a and b be elements of order 2 of a finite group G . Prove that 〈a ,b 〉 is a dihedral
group.
10. Find all subgroups of D12. Which of these are normal subgroups?
11. Show that GL(2, 2) ∼= S3.
12. What is the largest order of an element of S12?
13. Give an example of two non-isomorphic groups whose automorphism groups are
isomorphic.
14. If G is a non-cyclic abelian group, then AutG is non-abelian.
15. Let G act transitively on a set X , let H be a subgroup of G , and choose x ∈ X . Prove
that the following are equivalent:
(a) G = H stabG (x ),
(b) G = stabG (x )H ,
(c) H acts transitively on X .
Use this to find an alternative proof of Frattini’s argument.
16. Let H be a subgroup of G . Show that NG (H )/CG (H ) is isomorphic to a subgroup of
Aut H .
17. Find the center and all conjugacy classes of D2n .
18. Let P be a Sylow p -subgroup of a finite group G . Prove that if N is a normal
subgroup of G , then P ∩ N is a Sylow p -subgroup of N , and PN /N is a Sylow p -
subgroup of G /N .
19. Let P be a Sylow p -subgroup of a finite group G and H ≤ G . Is it true that P ∩ H is
always a Sylow p -subgroup of H ?
2. Let H be a subgroup of a group G with |G : H | = 2. Prove that H is a normal sub-
group of G .
3. Is it always true that if H is a subgroup of G with prime index, then H G ?
4. Let p be the smallest prime that divides the order of a finite group G . If H is a
subgroup of G of index p , then H is normal in G .
5. Find a group G and subgroups H and K with the property that H K G , but H is
not normal in G .
6. Let H and K be subgroups of finite index in G . Prove that |G : H ∩K | ≤ |G : H |·|G : K |,
with equality if and only if G = H K .
7. If H is a subgroup of G of finite index, then H contains a subgroup of finite index
which is normal in G .
8. A group in which every non-trivial element has order 2 is abelian.
9. Let a and b be elements of order 2 of a finite group G . Prove that 〈a ,b 〉 is a dihedral
group.
10. Find all subgroups of D12. Which of these are normal subgroups?
11. Show that GL(2, 2) ∼= S3.
12. What is the largest order of an element of S12?
13. Give an example of two non-isomorphic groups whose automorphism groups are
isomorphic.
14. If G is a non-cyclic abelian group, then AutG is non-abelian.
15. Let G act transitively on a set X , let H be a subgroup of G , and choose x ∈ X . Prove
that the following are equivalent:
(a) G = H stabG (x ),
(b) G = stabG (x )H ,
(c) H acts transitively on X .
Use this to find an alternative proof of Frattini’s argument.
16. Let H be a subgroup of G . Show that NG (H )/CG (H ) is isomorphic to a subgroup of
Aut H .
17. Find the center and all conjugacy classes of D2n .
18. Let P be a Sylow p -subgroup of a finite group G . Prove that if N is a normal
subgroup of G , then P ∩ N is a Sylow p -subgroup of N , and PN /N is a Sylow p -
subgroup of G /N .
19. Let P be a Sylow p -subgroup of a finite group G and H ≤ G . Is it true that P ∩ H is
always a Sylow p -subgroup of H ?
