List of top Mathematics Questions on Group Theory

Let \( G \) be the power set of \( \{1, 2, 3, 4\} \). Let the binary operation \( \Delta \) on \( G \) be

Which of the following statements are false?

Which of the following statements is false?
(I) \(\mathbb{Z}_5\times\mathbb{Z}_7\) is a cyclic group.
(II) The order of the element \((2,5)\in\mathbb{Z}_5\times\mathbb{Z}_7\) is 10.
(III) \(\text{Aut}(\mathbb{Z}_5\times\mathbb{Z}_7)\) is isomorphic to \(\mathbb{Z}_{35}\).

Which one of the following options is correct? (\(p\) is a prime number.)

If \(G\) is a group of order 30, then the number of Sylow 5-subgroups in \(G\) must be ____.

In a group \(G\), let \(x^5=e\) (\(e\): identity element of \(G\)) and \(xyx^{-1}=y^2\) for \(x,y\in G\). If \(y\ne e\), then the order of \(y\) is ____.

Let \(G\) be a finite Abelian group. If the subgroups \(H\) and \(K\) are of index 3 each in \(G\), then what is the index of the subgroup \(H\cap K\) in \(G\)?

Which of the following statements is/are true?
(I) The maximum possible order of an element in the group \(S_5\) (with composition \(\circ\)) is 6.
(II) If \(f:(S_3,\circ)\to(\mathbb{Z}_6,+_6)\) is a group homomorphism, then the order of \(f(S_3)\) is 1, 2 or 3.

Which one of the following statements is incorrect?
(I) The group \(\mathbb{Z}_{15}\) is isomorphic to \(\mathbb{Z}_3\times\mathbb{Z}_5\)
(II) \(\text{Inn}(A_3)\) is isomorphic to \(A_3\)

Which of the following statements is/are correct?
(I) In \(GL_2(\mathbb{R})\), matrices with the same determinant always belong to the same conjugacy class.
(II) The class equation of \(A_5\) (the alternating group on 5 elements) is \(60=1+6+10+15+28\).

In the dihedral group \(D_5=\{r,s:\; r^5=e,\; s^2=e,\; sr=r^{-1}s\}\) under composition as the binary operation, which of the following options is NOT true?