fix 4. Prove that H is a subgroup of G. Also prove (7) Let H≤ G, then the set aHa is a subgroup for each a € G, and HaHa-¹. TC

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PROBLEMS IN GROUP THEORY: NORMAL SUBGROUPS
Dr. A Sacidi
Group structures, University of Limpopo, 2022
In Questions 1-10 below, G is a finite group and e is the identity element.
(1) Prove that Z(G) is a normal subgroup of G.
(2) We know that if HKG and HG, then H4 K. Give an
example of a group with HAK4G but H is not normal in G.
(3) Let N and K be normal subgroups of G. Prove that NOK is also
normal in G.
(4) Let N and H be subgroups of G and let N 4G. Prove that NH
is normal in H.
(5) If N and K are normal in G with Kn N = {e}, then nk
all ke K and n E N.
=
(6) Let H be a subgroup of S4 consist of all those permutations o that
fix 4. Prove that H is a subgroup of G. Also prove that H = S3.
kn for
1
(7) Let H≤ G, then the set aHa is a subgroup for each a € G, and
HaHa-¹.
1
(8) Let G be a finite group and H a subgroup of G of order n. If H is
the only subgroup of G of order n, then H is normal in G.
(9) Let N and K be normal subgroups of G. Show that the subgroup
generated by HUK is also normal in G.
This one is hard.
1
(10)
¹ Let H and N be subgroup of a finite group G and suppose that
N is normal in G. Prove that if |G:H and N are relatively prime,
then H<N.
1
Transcribed Image Text:PROBLEMS IN GROUP THEORY: NORMAL SUBGROUPS Dr. A Sacidi Group structures, University of Limpopo, 2022 In Questions 1-10 below, G is a finite group and e is the identity element. (1) Prove that Z(G) is a normal subgroup of G. (2) We know that if HKG and HG, then H4 K. Give an example of a group with HAK4G but H is not normal in G. (3) Let N and K be normal subgroups of G. Prove that NOK is also normal in G. (4) Let N and H be subgroups of G and let N 4G. Prove that NH is normal in H. (5) If N and K are normal in G with Kn N = {e}, then nk all ke K and n E N. = (6) Let H be a subgroup of S4 consist of all those permutations o that fix 4. Prove that H is a subgroup of G. Also prove that H = S3. kn for 1 (7) Let H≤ G, then the set aHa is a subgroup for each a € G, and HaHa-¹. 1 (8) Let G be a finite group and H a subgroup of G of order n. If H is the only subgroup of G of order n, then H is normal in G. (9) Let N and K be normal subgroups of G. Show that the subgroup generated by HUK is also normal in G. This one is hard. 1 (10) ¹ Let H and N be subgroup of a finite group G and suppose that N is normal in G. Prove that if |G:H and N are relatively prime, then H<N. 1
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