Abstract Algebra Answers

Solve the equation 3𝑥 = 2 𝑖𝑛 𝑍7 𝑎𝑛𝑑 𝑖𝑛 𝑍23

compute the factor group (𝑍2 × 𝑍4)/< (1,2) >

Find the order of the following

(𝑍2 × 𝑍4)/< (1,1) >

Question

For sets H and K, we define the intersection H n K by H n K = {xI

X E H and X € K%. Show that if H ≤ G and K < G, then H n K < G.

(Remember: ≤ denotes "is a subgroup of," not "is a subset of.")

For each binary operation ∗ defined below say whether the following is

a group or not

a) Define ∗ 𝑜𝑛 𝑍 by 𝑎 ∗ 𝑏 = 𝑎 − 𝑏

b) Define ∗ 𝑜𝑛 𝑍 by 𝑎 ∗ 𝑏 = 𝑎𝑏

c) Define ∗ 𝑜𝑛 𝑅+by 𝑎 ∗ 𝑏 = 𝑎𝑏

d) Define ∗ 𝑜𝑛 𝑄 by 𝑎 ∗ 𝑏 = 𝑎𝑏

In (𝑍, +) , let 𝐻 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 3 and

𝐾 = 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 5.

Show that H and K are subgroups of Z . Also describe 𝐻 ∩ 𝐾

Determine whether the given set of invertible nXn matrices with real number entries is a subgroup of GL (n, R). The nXn matrices with determinant -1 or 1

Prove that every non-trivial sub group of cyclic group has finite index. Hence prove that (Q,+) is not cyclic.