2.3 Subgroups

1.
a. No, 1 + 1 ∉ H.
c. No, 32 = 9 ∉ H.
e. No, (1 2)(3 4) · (1 3)(2 4) = (1 4)(2 3) ∉ H.
g. Yes, 0 = 6 img H. H is closed because it consists of the even residues in img; −4 = 2, −2 = 4, so it is closed under inverses.
i. Yes, the unity (0, 0) img H. If (m, k) and (m′, k′) are in H, then so is (m, k) + (m′, k′) = (m + m′, k + k′) and −(m, k) = (− m, − k).
3. Yes. If H is a subgroup of G and K is a subgroup of H, then 1 img K (it is the unity of H). If a, b img K, then ab img K because this is their product in H. Finally, a−1 is the inverse of a in H, hence in K.
5. a. We have 1 img H because 1 = 12. If a, b img H, then a−1 = a (because a2 = 1), so a−1 H. Finally, the fact that ab = ba gives (

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