Abstract Algebra: Cosets Lagranges Theorem and Normal Subgroups

Let's examine the difference between a left coset and a right coset using the group $S_4$ and a subgroup $H$. Compute the left cosets and right cosets of a specific element and show that they are generally different in a non-abelian group.

Given: $\phi$ is a homomorphism from $G$ to $H$, $H$ is an abelian group, and the kernel of $\phi$ is a subset of $N$, where $N$ is a subgroup of $G$. Prove that $N$ is a normal subgroup of $G$.

Prove the two properties: 1) If $aH = bH$, then $a \in bH$. 2) If $a \in bH$, then $aH = bH$.

Let G be the group of integers under addition and H be the subgroup 3ℤ. What are the cosets of H in G? Construct the Cayley table for the factor group G/H.

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?

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.