New York University Non Trivial Group Homomorphism Modern Algebra Exam

total of 9 questions please read the details

1. 2.

Let n 2 N, let G be a group of order 2n, and let H be a subgroup of order n. Prove that gH = Hg for all g 2 G.

(i) List all the permutations in A4. (ii) Let

H = {1, (12)(34), (13)(24), (14)(23)}be a subgroup of A4. Find the left cosets of H, and the right cosets of H.

(iii) Using your results from part (ii), decide if H is a normal subgroup of A4. (iv) Let

K = {1, (234), (243)}be a subgroup of A4. Find the left cosets of K, and the right cosets of K.

(v) Using your results from part (iv), decide if K is a normal subgroup of A4. (vi) Calculate [A4 : H] and [A4 : K].

Let ‘: G ! H be a group homomorphism. Let g 2 G be an element of finite order.

1. (i) Prove that ‘(g) has finite order in H, and show that the order of ‘(g) divides the order of g.
2. (ii) Prove that the order of ‘(g) is equal to the order of g if ‘ is an isomorphism.
3. (iii) By considering elements of order 2, explain why D6 is not isomorphic to A4.

Let ‘: G ! H be a non-trivial group homomorphism. Suppose that |G| = 42 and |H| = 35.

(i) What is the order of ker ‘?(ii) What is the order of the image of ‘?

(i) Solve the congruence 7x ⌘ 13 mod 11. (ii) Solve the equation 6x = 17 in F19.

(iii) How many solutions does the equation 6x = 5 have in Z/9Z?

For references, consult §VIII–XV of the lecture notes or §2.5–2.11, 3.2 of Artin.
1. Let n 2 N, let G be a group of order 2n, and let H be a subgroup of order n.
Prove that gH = Hg for all g 2 G.
2.
(i) List all the permutations in A4 .
(ii) Let
H = {1, (12)(34), (13)(24), (14)(23)}
be a subgroup of A4 . Find the left cosets of H, and the right cosets of H.
(iii) Using your results from part (ii), decide if H is a normal subgroup of A4 .
(iv) Let
K = {1, (234), (243)}
be a subgroup of A4 . Find the left cosets of K, and the right cosets of K.
(v) Using your results from part (iv), decide if K is a normal subgroup of A4 .
(vi) Calculate [A4 : H] and [A4 : K].
3. Let ‘ : G ! H be a group homomorphism. Let g 2 G be an element of finite
order.
(i) Prove that ‘(g) has finite order in H, and show that the order of ‘(g)
divides the order of g.
(ii) Prove that the order of ‘(g) is equal to the order of g if ‘ is an isomorphism.
(iii) By considering elements of order 2, explain why D6 is not isomorphic to A4 .
4. Let ‘ : G ! H be a non-trivial group homomorphism. Suppose that |G| = 42
and |H| = 35.
(i) What is the order of ker ‘?
(ii) What is the order of the image of ‘?
5.
(i) Solve the congruence 7x ⌘
13 mod 11.
(ii) Solve the equation 6x = 17 in F19 .
(iii) How many solutions does the equation 6x = 5 have in Z/9Z?
1
6. Let G be a group and let ‘ : G ! G be the map defined by ‘(x) = x 1 for each
x 2 G. Prove that ‘ is an automorphism of G if and only if G is abelian.
7. Let C2 = hxi and C6 = hyi be cyclic groups of order 2 and 6 respectively. Let
G = C2 ⇥ C6 .
(i) List all the elements in G.
(ii) Is G a cyclic group?
(iii) Explain why H = h(1, y 2 )i is a normal subgroup of G.
(iv) Find the cosets of H.
8.
(i) List all 24 elements of the group SL2 (F3 ).
(ii) Let
C=
⇢

1 0
2 0
,
0 1
0 2
.
Prove that C is a normal subgroup of SL2 (F3 ).
(iii) Calculate [SL2 (F3 ) : C]?
(iv) According to Lagrange’s theorem, what are the possible orders of subgroups
of SL2 (F3 )?
(v) Find 2 proper subgroups of SL2 (F3 ) which have di↵erent orders, and with
order at least 3.
(vi) What is the order of GL2 (F3 )?
9. Let ‘ : G ! H be a group homomorphism. Let
L = {(x, ‘(x)) | x 2 G}.
Prove that L is a subgroup of G ⇥ H.
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