Let o(G) be the average order of a finite group G. We show that if o(G)<c, where c∈{136,114}, then G is an elementary abelian 2-group or solvable group, respectively. Also, we prove set containing orders all groups not dense in [a,∞), for a∈[0,136]. also outline some results related to integer values order. Since element popular research topic, pose open problems concerning throughout paper.

We present the first constant-round, tree-based, group key exchange protocol based on SIDH with logarithmic-order communication and memory complexity, where only previous isogeny-based exchange, SIBD, has linear-order complexity. call our supersingular isogeny tree-based (SIT). show that satisfies post-quantum security through a reduction to decisional Diffie–Hellman (SSDDH) problem in model of...