Free-cutting steel includes a specific composition, with an A value defined by a following formula (1) satisfying 4.0 or more and 20.0 or less, and remainder having Fe and an unavoidable impurity; and texture with 1000 or more sulfide particles with an equivalent circle diameter of less than 1 μm per mm.sup.2, 500 or more sulfide particles with an equivalent circle diameter of 1 μm or more and 5 μm or less per mm.sup.2, and 1000 or more Pb particles with an equivalent circle diameter of 1 μm or less per mm.sup.2, wherein

A value=(Mn+5Cr)/S   (1)

here, symbols of elements in the formula indicate contents (% by mass) of the elements.

Free-cutting steel includes a specific composition, with an A value defined by a following formula (1) satisfying 4.0 or more and 20.0 or less, and remainder having Fe and an unavoidable impurity; and texture with 1000 or more sulfide particles with an equivalent circle diameter of less than 1 μm per mm.sup.2, 500 or more sulfide particles with an equivalent circle diameter of 1 μm or more and 5 μm or less per mm.sup.2, and 1000 or more Pb particles with an equivalent circle diameter of 1 μm or less per mm.sup.2, wherein

A value=(Mn+5Cr)/S   (1)

here, symbols of elements in the formula indicate contents (% by mass) of the elements.

A steel sheet has a predetermined chemical composition containing, in mass %, C: more than 0.18% and less than 0.30%, Mn: more than 2.50% and 4.00% or less, and other elements. The steel micro-structure at a position at a depth of ¼ of the sheet thickness from the surface in an L cross section of the steel sheet is, in area %, tempered martensite: 25 to 90%, ferrite: 5% or less, retained austenite: 10 to 50%, and bainite: 5% or less. At a position at a depth of ¼ of the sheet thickness from the surface of the L cross section, the proportion of a total area of retained austenite grains having an area of 1 μm.sup.2 or more and having a grain shape circularity of 0.1 or more is less than 50% with respect to the entire area of retained austenite. The steel sheet satisfies the formula C.sub.Mnγ/C.sub.Mnα≥1.2.

Provided is a grain-oriented electrical steel sheet that has excellent magnetic properties and can be manufactured by secondary recrystallization orientation control using coil annealing with high productivity. A grain-oriented electrical steel sheet comprises a specific chemical composition, wherein an average value of a deviation angle (α.sup.2+β.sup.2).sup.1/2 calculated from a deviation angle α from ideal Goss orientation around an ND rotation axis and a deviation angle β from ideal Goss orientation around a TD rotation axis is 5.0° or less, and an area ratio R.sub.β of crystal grains with β≤0.50° is 20% or less.

Provided is a grain-oriented electrical steel sheet that has excellent magnetic properties and can be manufactured by secondary recrystallization orientation control using coil annealing with high productivity. A grain-oriented electrical steel sheet comprises a specific chemical composition, wherein an average value of a deviation angle (α.sup.2+β.sup.2).sup.1/2 calculated from a deviation angle α from ideal Goss orientation around an ND rotation axis and a deviation angle β from ideal Goss orientation around a TD rotation axis is 5.0° or less, and an area ratio R.sub.β of crystal grains with β≤0.50° is 20% or less.

A grain-oriented electrical steel sheet includes: a base steel sheet having a predetermined chemical composition; a glass coating provided on the surface of the base steel sheet; and a tension-applying insulation coating provided on the surface of the glass coating, in which linear thermal strains having, a predetermined angle (φ) with respect to a transverse direction which is a direction orthogonal to a rolling direction are periodically formed on the surface of the tension-applying insulation coating at predetermined intervals along the rolling direction, a full width at half maximum F1 on the linear thermal strain and a full width at half maximum F2 at an intermediate position between the two linear thermal strains adjacent to each other satisfy 0.00<(F1−F2)/F2≤0.15, the width of the linear thermal strain is 10 μm or more and 300 μm or less, and in the base steel sheet, an orientation distribution angle γ around a rolling direction axis of secondary recrystallization grains, an orientation distribution angle α around an axis parallel to a normal direction, and an orientation distribution angle β around an axis perpendicular to each of the RD axis and the ND axis in units of ° satisfy 1.0≤γ≤8.0 and 0.0≤(α.sup.2+β.sup.2).sup.0.5≤10.0.

A grain-oriented electrical steel sheet includes: a base steel sheet having a predetermined chemical composition; a glass coating provided on the surface of the base steel sheet; and a tension-applying insulation coating provided on the surface of the glass coating, in which linear thermal strains having, a predetermined angle (φ) with respect to a transverse direction which is a direction orthogonal to a rolling direction are periodically formed on the surface of the tension-applying insulation coating at predetermined intervals along the rolling direction, a full width at half maximum F1 on the linear thermal strain and a full width at half maximum F2 at an intermediate position between the two linear thermal strains adjacent to each other satisfy 0.00<(F1−F2)/F2≤0.15, the width of the linear thermal strain is 10 μm or more and 300 μm or less, and in the base steel sheet, an orientation distribution angle γ around a rolling direction axis of secondary recrystallization grains, an orientation distribution angle α around an axis parallel to a normal direction, and an orientation distribution angle β around an axis perpendicular to each of the RD axis and the ND axis in units of ° satisfy 1.0≤γ≤8.0 and 0.0≤(α.sup.2+β.sup.2).sup.0.5≤10.0.

A high-strength galvanized steel sheet includes a steel sheet having a chemical composition containing a predetermined component element, a mass ratio of a content of Si to a content of Mn in the steel (Si/Mn) being 0.1 or more and less than 0.2, and the balance: Fe and incidental impurities, and a steel structure in which an average grain size of inclusions containing at least one of Al, Si, Mg, and Ca and existing in an area extending from a surface to a position of ⅓ of a sheet thickness is 50 μm or less, and an average nearest distance between ones of the inclusions is 20 μm or more; and a galvanized layer provided on a surface of the steel sheet, in which an amount of diffusible hydrogen contained in the steel is less than 0.25 mass ppm, and a tensile strength is 1100 MPa or more.

A case hardening steel having excellent fatigue resistance is provided at relatively low production cost. A case hardening steel has a chemical composition containing C: 0.10% to 0.30%, Si: 0.10% to 1.20%, Mn: 0.30% to 1.50%, S: 0.010% to 0.030%, Cr: 0.10% to 1.00%, B: 0.0005% to 0.0050%, Sb: 0.005% to 0.020%, and N: 0.0150% or less in a predetermined range, and further containing Al: 0.010%≤Al≤0.120% in the case where B−(10.8/14)N≥0.0003%, and 27/14[N−(14/10.8)B+0.030]≤Al≤0.120% in the case where B−(10.8/14)N<0.0003%.

A case hardening steel having excellent fatigue resistance is provided at relatively low production cost. A case hardening steel has a chemical composition containing C: 0.10% to 0.30%, Si: 0.10% to 1.20%, Mn: 0.30% to 1.50%, S: 0.010% to 0.030%, Cr: 0.10% to 1.00%, B: 0.0005% to 0.0050%, Sb: 0.005% to 0.020%, and N: 0.0150% or less in a predetermined range, and further containing Al: 0.010%≤Al≤0.120% in the case where B−(10.8/14)N≥0.0003%, and 27/14[N−(14/10.8)B+0.030]≤Al≤0.120% in the case where B−(10.8/14)N<0.0003%.