Question

there exists a non-cyclic group in which every proper subgroup is cyclic.true or false prove

Answer 1

Answer on Question #64367 – Math – Abstract Algebra

Question

There exists a non-cyclic group in which every proper subgroup is cyclic. True or False. Prove.

Solution

Consider the dihedral group $D_3$, that is the symmetry group of an equilateral triangle. The multiplication table of this group is given below:

From the table we see that $G$ is non-abelian, because the table is not symmetric. Therefore, $G$ is non-cyclic. The proper nonempty subsets, which contain $r_0$ and closed under multiplication, are:

$$\begin{array}{l} S_1 = \{r_0\}, \\ S_2 = \{r_0, r_1, r_2\}, \\ S_3 = \{r_0, s_1\}, \\ S_4 = \{r_0, s_2\}, \\ S_5 = \{r_0, s_3\} \end{array}$$

The subsets $S_1, S_3, S_4$ and $S_5$ are cyclic subgroups because the elements $r_0, s_1, s_2$ and $s_3$ are their own inverses. The subset $S_2$ is a cyclic subgroup because $r_1$ is a generator for it.

Answer: True.

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