NOVELTY - A positive-electrode active material particle powder consists of a compound having a crystal system which belongs to space group R-3m at least, and a crystal system which belongs to space groups C2/m. The compound consists of a complex oxide containing lithium, manganese and cobalt, and/or nickel, element (A) chosen from aluminum, magnesium, titanium, zirconium, calcium, iron, zinc, molybdenum and bismuth, boron, and fluorine. The molar ratio of manganese with respect to total content of nickel, cobalt and manganese is more than 0.55.    USE - Active material particle powder is used for positive electrodes of nonaqueous electrolyte secondary batteries (claimed) e.g. lithium ion batteries used for motor vehicles, audio-video equipment, and personal computers.    ADVANTAGE - The positive-electrode active material has excellent charge/discharge capacity, cycle characteristics, and thermal stability.    DETAILED DESCRIPTION - A positive-electrode active material particle powder consists of a compound having a crystal system which belongs to space group R-3m at least, and a crystal system which belongs to space groups C2/m. The compound consists of a complex oxide containing lithium, manganese and cobalt, and/or nickel, element (A) chosen from aluminum, magnesium, titanium, zirconium, calcium, iron, zinc, molybdenum and bismuth, boron, and fluorine. The molar ratio of manganese with respect to total content of nickel, cobalt and manganese is more than 0.55. The content of boron is 0.01-1 wt.%, the content of fluorine is 0.01-5 wt.%, and the content of element (A) is 0.004-9 wt.%. The ratio of intensity of the maximum diffraction peak in 2 theta of 20.8 plus minus 1 degrees of powder X-ray diffraction pattern which uses Cu-K- alpha X-ray of the positive-electrode active material particle powder, and intensity of the maximum diffraction peak in 2 theta of 18.6 plus minus 1 degrees is 0.02-0.2. An INDEPENDENT CLAIM is included for manufacture of positive-electrode active material particle powder, which involves baking a mixture obtained by dry-mixing precursor particle powder containing transition metal, boron compound, lithium compound, fluorine compound, and compound of element (A), mixing and baking lithium compound to dried powder of mixture wet-blended with boron compound, fluorine compound, and compound of element (A) to slurry containing precursor particles containing transition metal, or baking dry type or mixture powder of precursor particle powder, boron compound, and lithium compound containing transition metal, or heat-processing fluorine compound and compound of element (A) in form of dry type or mixture powder.