• Deepak Gumber

      Articles written in Proceedings – Mathematical Sciences

    • A Note on Conjugacy Classes of Finite Groups

      Hemant Kalra Deepak Gumber

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      Let 𝐺 be a finite group and let $x^G$ denote the conjugacy class of an element 𝑥 of 𝐺. We classify all finite groups 𝐺 in the following three cases: (i) Each non-trivial conjugacy class of 𝐺 together with the identity element 1 is a subgroup of 𝐺, (ii) union of any two distinct non-trivial conjugacy classes of 𝐺 together with 1 is a subgroup of 𝐺, and (iii) union of any three distinct non-trivial conjugacy classes of 𝐺 together with 1 is a subgroup of 𝐺.

    • Class-preserving automorphisms of some finite 𝑝-groups

      Deepak Gumber Mahak Sharma

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      Let 𝐺 be a finite 𝑝-group of order 𝑝5, where 𝑝 is a prime. We give necessary and sufficient conditions on 𝐺 such that 𝐺 has a non-inner class-preserving automorphism. As a consequence, we give short and alternate proofs of results of Yadav (§5 of Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008) 1–11) and Kalra and Gumber (Theorem 4.2 of Indian J. Pure Appl. Math. 44 (2013) 711–725).

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