Abstract Algebra

Show that the inverse of an inverse is itself.

Every element of a group has exactly one inverse.

Let G be a finite group and H a subgroup of G. Determine the possible orders of the subgroups of G based on Lagrange's Theorem.

If a group G has 12 elements, what are the possible orders for its subgroups according to Lagrange's Theorem?

If a group G has 7 elements, what are the possible orders for its subgroups?

Given a finite group G and a subgroup H, demonstrate Lagrange's Theorem by showing that the number of left cosets of H in G divides the order of G.

Let G be $\mathbb{Z}_6$ and H be the set {0, 3}. Are the left cosets and the right cosets the same, making H a normal subgroup of G?

Let G be $S_3$ and H be the subgroup {identity, (1 2 3), (1 3 2)}. Show that H is a normal subgroup of G by proving that each left coset is the same as the right coset.

Let G be $S_3$ and H be the set {identity, (1 2)}. Determine if H is a normal subgroup by checking if the left and right cosets are not the same.

Prove that the number of elements in the alternating group $A_n$ is $\frac{n!}{2}$.

In the alternating group $A_{10}$, show that there is an element of order 15.

Is a subset S of a ring R a subring of R under the operations of R?

A Boolean ring is any ring where for any element $x$, $x^2 = x$. Prove the following: 1. For all elements $x$ in a Boolean ring, $x + x = 0$. 2. Every Boolean ring is commutative, i.e., for all elements $x$ and $y$ in a Boolean ring, $xy = yx$.

Prove that the mapping from the integers $\mathbb{Z}$ into a ring with unity, where an integer $n$ maps to $n \times 1$, preserves addition and multiplication, and therefore is a homomorphism.

If a ring has a positive characteristic $n$, show that it contains a subring isomorphic to $\mathbb{Z}_n$.

For a field of characteristic $p$, show that it contains a subfield isomorphic to $\mathbb{Z}_p$.