Monoids and Groups for GATE

Which of the following option is True?

If (G, *) is a cyclic group of order 480 then find the number of generators in G?

The set {1, 2, 4, 7, 8, 11, 13, 14} is a group under multiplication modulo 15. The inverses of 4 and 7 are respectively:

For the composition table of a cyclic group shown below:

[Tex]\begin{array}{|c|c|c|c|c|} \hline \textbf{*} & \textbf{a}& \textbf{b} &\textbf{c} & \textbf{d}\\\hline \textbf{a} & \text{a}& \text{b} & \text{c} & \text{d} \\\hline \textbf{b} & \text{b}& \text{a} & \text{d} &\text{c}\\\hline \textbf{c} & \text{c}& \text{d} & \text{b} & \text{a}\\\hline \textbf{d} & \text{d}& \text{c} & \text{a} & \text{b} \\\hline \end{array}[/Tex]

Which one of the following choices is correct?

Which one of the following is NOT necessarily a property of a Group?

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is _______.

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is ________.

How many non-isomorphic abelian groups of order 64 are there?

Let G be a cyclic group of order 60. How many subgroups does G have?