List questions from cyclic-groups

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List Question

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Find homomorphisms from $C_8$ to $S_3$ and from $C_6$ to $S_3$, and are there any isomorphisms?

Subgroup of a cyclic groups and are isomorphic

Prove that $\sigma_a \circ \sigma_b = \sigma_{ab}$.

Proof blueprint - If $G/Z(G)$ cyclic then $G$ Abelian - Fraleigh p. 153 15.37

Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

If a group has only one element $a$ of order $n$, then $a$ belongs to $Z (G)$ and $n=2$.

Cyclic subgroup of $S_n$

Regarding the order of elements in a factor group

Order of ab when <a> and <b> are distinct

Finding a permutation $ \alpha $ given $ \alpha^4 $

G is group of order pq, pq are primes

Abstract Algebra: Cosets

Lagrange's Theorem Proof Help

Proper subgroup is cyclic if $|G|=p^2$

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