Question

Let G = {e, a, a^2, a^3} be the cyclic group of order 4, and consider the subgroup H = {e, a^2}. Compute the left cosets of H and the index [G : H]. (2) Consider the symmetric group S4 and the cyclic subgroup H generated by the cycle (1, 2, 3, 4). Compute the left cosets of H and the index [S4 : H]. (3) Prove that if H1 and H2 are subgroups of a group G, then H1 ? H2 is a subgroup of G. (4) (a) Prove that the group S3 contains no subgroup of order 4. (b) Prove that the group S4 contains no subgroup of order 5. (5) Consider the additive group R and the subgroup Z. Prove that R/Z = {x + Z : 0 ? x < 1} and that (x + Z) ? (y + Z) = ? if 0 ? x < y < 1. (6) In the group S5, consider the permutation ? = (1 2 3 4 5) (2 3 1 5 4).

          Let G = {e, a, a^2, a^3} be the cyclic group of order 4, and consider the subgroup H = {e, a^2}. Compute the left cosets of H and the index [G : H].

(2) Consider the symmetric group S4 and the cyclic subgroup H generated by the cycle (1, 2, 3, 4). Compute the left cosets of H and the index [S4 : H].

(3) Prove that if H1 and H2 are subgroups of a group G, then H1 ? H2 is a subgroup of G.

(4) (a) Prove that the group S3 contains no subgroup of order 4.
(b) Prove that the group S4 contains no subgroup of order 5.

(5) Consider the additive group R and the subgroup Z. Prove that

R/Z = {x + Z : 0 ? x < 1}

and that

(x + Z) ? (y + Z) = ? if 0 ? x < y < 1.

(6) In the group S5, consider the permutation

? = (1 2 3 4 5)
    (2 3 1 5 4).

Let G = e, a, a^2, a^3 be the cyclic group of order 4, and consider the subgroup H = e, a^2. Compute the left cosets of H and the index [G : H].

(2) Consider the symmetric group S4 and the cyclic subgroup H generated by the cycle (1, 2, 3, 4). Compute the left cosets of H and the index [S4 : H].

(3) Prove that if H1 and H2 are subgroups of a group G, then H1 ? H2 is a subgroup of G.

(4) (a) Prove that the group S3 contains no subgroup of order 4.
(b) Prove that the group S4 contains no subgroup of order 5.

(5) Consider the additive group R and the subgroup Z. Prove that

R/Z = x + Z : 0 ? x < 1

and that

(x + Z) ? (y + Z) = ? if 0 ? x < y < 1.

(6) In the group S5, consider the permutation

? = (1 2 3 4 5)
(2 3 1 5 4).

Added by Troy L.

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