Chapter 1

Show that an element of a group has exactly one inverse.

Suppose that \(H\) is a subgroup of \(G\) of index 2 . Show that \(H\) is a normal subgroup of \(G\)

Suppose that \(G\) has exactly one subgroup \(H\) of order \(k\). Show that \(H\) is a normal subgroup of \(G\)

Suppose that \(H\) is a normal subgroup of \(G\) and that \(K\) is a normal subgroup of \(H\). Is \(K\) necessarily a normal subgroup of \(G ?\)

Show that any permutation of a finite set can be written as a product of transpositions.

Show that a group \(G\) is generated by each of its elements (other than the unit element) if and only if \(G\) is a finite cyclic group of prime order.

Describe the elements of \(\mathbb{Z}_{n}\) which generate \(\mathbb{Z}_{n}\) (for any positive integer \(n\) ).

If \(\pi \in \Sigma_{n}\), let \(\varepsilon(\pi)=\prod_{i

Every field has at least two elements. Show that there is a field with exactly two elements.

Which of the following subsets of \(\mathbb{C}\) are subfields of \(C\) ? (i) \(\\{a+i b: a, b \in \mathbb{Q}\\}\). (ii) \(\left\\{a+\omega b: a, b \in \mathbb{Q}, \omega=\frac{1}{2}(-1+\sqrt{3} \mathrm{i})\right\\}\) (iii) \(\left\\{a+2^{1 / 3} b: a, b \in \mathbb{Q}\right\\}\). (iv) \(\left\\{a+2^{1 / 3} b+4^{1 / 3} c: a, b, c \in \mathbb{Q}\right\\}\).

Suppose that \(S\) is infinite. For each \(s\) in \(S\), let \(e_{s}(t)=1\) if \(s=t\), and let \(e_{s}(t)=0\) otherwise. Is \(\left\\{e_{s}: s \in S\right\\}\) a basis for \(K^{s} ?\)

\(\mathbb{R}\) can be considered as a vector space over \(\mathbb{Q}\). Show that \(\mathbb{R}\) is not finite dimensional over Q. Can you find an infinite subset of \(R\) which is linearly independent over \(Q\) ?

Suppose that \(K\) is an infinite field and that \(V\) is a vector space over \(K\). Show that it is not possible to write \(V=\bigcup_{i=1}^{n} U_{i}\), where \(U_{1}, \ldots, U_{n}\) are proper linear subspaces of \(V\)

Suppose that \(K\) is a finite field with \(k\) elements, and that \(V\) is an \(r\) dimensional vector space over \(K\). Show that if \(V=\bigcup_{i=1}^{n} U_{i}\), where \(U_{1}, \ldots, U_{n}\) are proper linear subspaces of \(V\), then \(n \geqslant\left(k^{r}-1\right) /(k-1)\) Show that there exist \(\left(k^{r}-1\right) /(k-1)\) proper linear subspaces of \(V\) whose union is \(V\).