1.1.06
March 27, 2019
Determine which of the following sets are groups under addition
1.1.12
March 29, 2019
Find the orders of the following elements of the multiplicative group \((\mathbb{Z}/12\mathbb{Z})^\times\)
1.1.15
March 29, 2019
Prove that \((a_1a_2...a_n)^{-1} = a_n^{-1}a_{n-1}^{-1}...a_1^{-1}\) for all \(a_1, a_2,...,a_n \in G\).
1.1.22
March 29, 2019
If \(x\) and \(g\) are elements of the group \(G\), prove that \(|x| = |g^{-1}xg|\). Deduce that \(|ab| = |ba|\) for all \(a, b \in G\).
1.1.25
March 30, 2019
Prove that if \(x^2 = 1 \) for all \(x \in G\) then \(G\) is abelian.
1.2.01
March 30, 2019
Compute the order of each of the elements in the following groups: \(D_6\), \(D_8\), \(D_{10}\)
1.2.02
March 30, 2019
Use the generators and relations above to show that if \(x\) is any element of \(D_{2n}\) which is not a power of \(r\), then \(rx=xr^{-1}\).
1.2.03
March 30, 2019
Use the generators and relations above to show that every element of \(D_{2n}\) which is not a power of \(r\) has order \(2\).
1.2.09
March 30, 2019
Let \(G\) be the group of rigid motions in \(\mathbb{R}^3\) of a tetrahedron. Show that \(|G| = 12\).
1.3.02
April 2, 2019
Find the cycle decompositions of the following permutations: \(\sigma, \tau, \sigma^2, \sigma\tau, \tau\sigma\) and \(\tau^2\sigma\).
1.3.05
April 2, 2019
Find the order of \((1\ 12\ 8\ 10\ 4)(2\ 13)(5\ 11\ 7)(6\ 9)\).
1.3.13
April 3, 2019
Show that an element has order \(2\) in \(S_n\) if and only if its cycle decomposition is a product of commuting \(2\)-cycles.
1.3.20
April 4, 2019
Find a set of generators and relations for \(S_3\).
1.4.07
April 13, 2019
Let \(p\) be a prime. Prove that the order of \(GL_2(\mathbb{F}_p)\) is \(p^4 - p^3 - p^2 + p\).
1.5.02
April 13, 2019
Write out the group tables for \(S_3, D_8\), and \(Q_8\).
1.6.01
April 14, 2019
Let \(\varphi:G\rightarrow H\) be a homomorphism. Prove that \(\varphi(x^n) = \varphi(x)^n\) for all \(n \in \mathbb{Z}^+\)...
1.7.11
April 15, 2019
Write out the cycle decomposition of the eight permutations in \(S_4\) corresponding to the elements of \(D_8\) given by the action of \(D_8\) on the vertices of a square.
1.7.17
April 17, 2019
Let \(G\) be a group and let each \(g\in G\) map \(G\) to \(G\) by \(x \mapsto gxg^{-1}\). For fixed \(g\in G\), prove that conjugation by \(g\) is an isomorphism from \(G\) onto itself.
1.7.18
April 17, 2019
Let \(H\) be a group acting on a set \(A\). Prove that the relation \(\sim\) on \(A\) defined by \(a \sim b \text{ if and only if } a = hb \text{ for some } h \in H\) is an equivalence relation.
1.7.19
April 19, 2019
Let \(H\) be a subgroup of the finite group \(G\) and let \(H\) act on \(G\) (here \(A = G\)) by left multiplication. Let \(x \in G\) and let \(O \) be the orbit of \(x\) under the action of \(H\). Prove that...